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I. A description of the meciulne .
We begin by describing the ’unsteckered enigma'. The machine
consists of a box with 26 keys labelled with the letters of the
alphabet and 26 bulbs whioh shine through stencils on whioh
letters are marked. It also cAntains wheelswhose funotion will
be described later on. When a key is depressed the wheels are
made to move in a oertain way and a ourrent flows through the
whearls to one of the bulbs. fgXBndtagalMnri xt tar The letter whioh
appears over the bulb is
the result of enciphering the
letter on the depres ed key with the wheels in the position they
have when the bulB lights .
To understand the working of the machine it Is best to separate
in our minds
ji* The electric oirouit of the machine without the wheels*
The oirouit through the who els.
* The meohenism ror turning the wheels and for describing
lithe positions of the wheels.
I H The oirouit of the machine without the wheel s .
4
Hi* Mtt u«6
L s-
1 1'
V; \ *
The meohine contains a cylinder called the Eintritt^swnlz
JlI.W)on whioh are 26 oonteots C x t C f The effeot of the
wheels is to oonneot these contacts uo in oairs, the actual
pairings of oodrse depending on the nos it ions of the wheels.
On the other side the oontects C t J C tm) i f C^ are oonneoted
eaoh to one of the keys. For the moment we wi 1 ! sup-ose that the
order id pgrt ncbuaiiigljjc ^WERTZU IOASDFGH^KPYXCV HNML , and we
will say that Q is the letter associated with Cj , W that associated
with C ^ etc. This series of letters associated with C%y ,
is called th e diagonal, for reasons which will appear in Chap
The particular order we have chosen is known as QWERTZU order.
The diagram shews the connections when the key Q is
depressed and supposin g that C ^ is connected to C ^ through the
The only outlet for the positive of the battery is through the Q
keyt o Cj hence to Cj and then through the E bulb • The result is
that the E bulb lights. More generally we can s°y
If two oontaots C , ^ of the Eintrittswalz ere oonneoted
throughthe wheels th en the result of enciphering the letter
- ' _ /
esseciated with C is the letter associated with C .
Notioe that if P is the result of enciphering G, then G is the
result of enciphering P at the same plaoe, also that the result
of enciphering G oan never be G.
Henoeforward we may negleot all of the maohine except th*
what affeots the connections between the contacts of the
E.W. , and the turnover* mechanism which affeots the nos it ions of
wheels.
Connections throughthe wheel s.
The wheels inolude one which is seldom removed from the
machine, and wh^ich may or may not be rotatable. It is oalled tbs
Umkeh^rwalz (U.K.W.). This wheel h as 26 spring oontaots which
are connected together in pairs. There are three or more other
wheels which are removable and- rots table ; they have 26 snring
contaots on theright end 26 plate oontaots on the left£ left and
right with saxmadc positions when in the meohinejt Each spring
3
oontaot is connected to one and only one nlate contaot. On the
wheels are rings or tyres carrying alphabets , and rotatable with
respect to the rest of the whe°l; more about this under ’turnovers*
When the maohine is being used three of the v/heels are nut in
between the U.K.W. amd the E,W. in some prescribed order . The
way that the ourrent might flow from the E.W. through th e wheels
and back is shewn below
L*.u «,M.w
UKV MV £.W
Turnovers. Rlngstellung. Wflndow -position . rofl^nosltlon .
From the point of view of the legitimate deoiohe-er, the
position of th e v/heels is described by the letters on the tyres
■fxtiuxnjnulE which shew through the thre^ (or 4 if the U.K.W.
rotates) windows in the oasing of the machine. This seauenoe of lette:
we oalllthe 'window position’ • When a key is depressed the window
when
position changes, but does not change further ara the key is
allowed to rise. We will say that the position ohanges into the
’following’ position. The position which follows a given one depends
only on the order of the wheels and on the original window
position. This is because the mechanism for changing the positions
is carried on the tyres .
The turning mechanism cohsists of
. w
Tharee palls operated by th e keys, one lying Just to the
right of th e right hand wheel, one between the R.H.W and M.W.
and one between the M.W. and th e L.H.W.
26 catches fixed on thacxxfcgicfc each wheel on the right .
One (or tntluDcoacKKXBf ros ibly more, here we will always
s'
assume it is only one) catch on each t.vrei^the left.
The effeot of the right hand pall is to move the xlgtetxha R.H.W.
forward one place every time a key is depressed. The middle pall
normally oomes into contact with the smooth surface of the tyre
which prevents
of the R.H.W. , ixexratiB?: it from nnndtKgx engaging with the
catohes of the M.W. If however it is able to slip in to the
oatoh on the tyre of the R.H.W. it will reach the oetch on the
M.W. 8nd will push both R.H.W. amd M.W. forward: of course the
R.H.W. is being pushed forward by the right hand pawl in any
oase. The occurence of suoh a movement of the M.W. is called a
* turnover'. Owing to the faot that the catch is on the tyre the
position at whioh the turnover ocoikrs depends only on what
wheel is in the right hand position, and on the window position
of that wheel. For instenoe with German servioe wheels , wheel I
turns over between Q and R , i.e. if I is in the R.H. position
then th e M.W. will move forward whenever the window positionof
the R.H.W. changes from Q to R. The left hand pawl operates
similarly to the middle pawl, but in this oase it is essential +o
remember that both M.W. and L.H.W. move forward.
Typical examples of oonseoutive window positions with middle
wheel jnm turnover E-F, EX R.H.W. T.O. Q -R
AWO
BDO
MEW
PE<i
AwP
BDP
NFS
v^FK
AWQ
BD^
NFY
AXR
nER
NFZ
(*FT
AXS
CFS
AST
CFT
The effect of enciphering a letter depends only on the
wheel order (Walzenlage) snd the position (i.e, amount rotated)
of the wheel proper (i.e. not the tyre). To desoribe this position
we could imagine that there was a set of letters attached to
the business part of eeoh wheel, and that these letters could
ac±xn,be seen throughthe windows as ell as the letters on the
tyres. The letters seen would give the ’absolute * or ’rod’ position
of the wheel (the point of th e expression ’rod position* will
be seen in Chap ). The position of the tyre relative to th e
business part is fixed by means of a clip on the business part
whioh oan drop into holes near the letters. When the dip ihsin thw
hole near the letter C v;e ssy that the Ringetellung is C for
that wheel. It is dear that sotoe equation of the form
Window position = Rod position Ringstellung -f- a constant
must hbld( it being understood that A,B,C,;.. are regarded
as interchangeable with 1,2,3,...). U£wxsjra±± normally m cTCKvmrtM wt
tins one arranges that this constant is zero (see also
The steckered enigma .
In some enigmas the association of the oon tacts of the
Eintrittswalz withthe keys and bulbs can be varied. There are 26
pairs of sockets labelled with the letters of the alphabet
one of eeoh pair leading to a oontaot of the Eintrittswalz and
the other to one of the keys. Normally the two sookets are
connected together by a hidden spring, if however a 'Steoker*
is plugged into two pairs of sookets, W anal R say, these springs
are foroed away and new connections ere made through the Steoker,
the W key being connected to the contact which would otherwise
be oonneoted to th e R key , and vice-versa. That W pnd R are
connected by such a plug is expres-ed in the form 'W/R* or *R/W* .
Th e effect of the Steoker on the encipherment is quite simple.
If at a oertain position of the out wheels A enciphered gives N,
(abbreviated to AN) then at the same position with Steoker
A/V,N/0, and perhaps others, we have VO; if instead we have the
Steoker A/V but none involving N , we should have VN (or as we
sometimes say the 'consta^ion' VN). Thus if a possible encipherment
without any Steoker were
maocismasHx; dieserbeabi
JMTKEYoHI
then a possible enoipherment starting from the same positions
of th e wheels (or as we say, from the s c me place) xei with the
Steoker D/S, E/N, B/E, T/y wouia be
SIEDENKE
BVMYBEVO
k'uM. /W “
r
Conventions for electricians
For the purpose of desribing the wiring of wheels to
electricians one works from a’spot* on the right hand (spring
contact bearing) side of the wheel, or if there is no spot^from
the contact which is uppermostwhen any writing on the face is
horizontal
The oontaot whioh is uppermost or nearest to the spot is called 1
and then the numbering is continued ina olookwise direction.
One then makes out a soheme like this
Spring contacts
Fixed contadts
1 2 3 4 5
63 16 14 .
*31
From the poin t of view of the oryptogranh er the most natural
way of naming th e oontacts is rather different. One would put
the Ringstellung to zero, then put zero (Z) in the window, and
after
name any oontaot on the right of the R.H.W. joy the letter
associated with the oontaot of the E.W. whioh it touches, there
being assumed to be no Stocker* To connect these two notations
it would be necessary to take into eonsideration the relative
positions of the dontact of the E.W. and the windows, and also
the positions of the clip and spot on the wheel. Here is a rule
of thumb for obtaining eleotrioians data from the oryntogranhio
data, illustrated by Railway Wheel I. W
Write down th e first upright of the inverse so up re for the whee
unsteokered
and above it the diagonal* Use the top two lines to* transpose' the
l l 3
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third line into numbers. Then rub out the seoond and third lines.
This rule is not raa absolutely reliable beoause of possible
variations of designs of wheels and maohines.
The comlo strips .
For demonstration purposes it is best to replace the maohine
by a paper model. We replace eeoh wheel by a nxpaiscxtxiLjD^xxnfxxx
strip of squared peper 52 squares by 5 squares. The squares in the
right hand column of the strip represent the spring oontacts of the
wheel in natural order (to make th e squares of the strip agree
with the contacts of the wheel one must wrap th e strip round the
wheel with th e writing on the strip inwards j. The squares on the
left represent the plate contacts. In the right hand column is writtei
the diagonal twice over, these being the ’cryptographers names' of th<
contacts as explained in the last section; in the left hand oolumn
letters are also written, and in such a way that squares containing
represent oontaots whioh are
the seme letter kxx connected together, Down th e centre oolumn
may be written the numbers 1, • . • ,26,1, . . . , 26. These numbers serve
to desoribe the position of the wheel, either the rod uosition or
the window position according to how they ere used. The Urakehrwwlz
is represented by e strip three squares wide, containing in one
oolumn the diagonal repeated (this is not entirely essential) in
another the numbers 1,...,26 repeatesd, The third oolumn represents
the oontaots; and squares representing oontaots whioh are connected
contain the same number (which does nor enoeed 13). The maohine itsel
is represented by a sheet of paper with slots to hold the 'wheels*.
In a column on the right is written the unsteokered diagonal twin
mr»T to represent the Eintrittswalz. It is convenient to repe-t
tfcumn this alphabet between eaoh pair of wheels. The square
bearing the letter Q between the R.H.W. and the M.W. will bar
called R.H.W. ’rod point Q» or M.W. ’output roint Q*. Between the
wheels we also write 1,...,26 repeated. These at*xx«xt«xd«atflrrtitte
tdasxSin* are used for describing the position of the wheel when
the Ringstellu ng is given. To understand how this oan be done
we need only notioe that the same effeot as a movable type
could be obtained by having windows and pawls which oould be
rotated roung the wheels in step. To use this Ringstellung device
on the oomlo strips we make penoil marks against the numbers
on th e fixed sheet and read off the window oositions on the strios
opposite these marks. We also make permanent lines on the strios
to shew where the turnove ooours. When these lines oass the
Ringstellung marks a turnover occurs.
If the machine has Steoker we may leave a column on the right
for the keys to which the contacts of the E.W. are oonneoted through
the Steoker.
The rule of thumb for the making of comic strips is to take
the lest upright of the rod squard for the left hand columns
of the strips.
I t may appear rather strange th n t the letters written on
the fixed shleet between th e strios should be in the order of
the diagonal, rather than say ABCD . . . • the ooint of -riting the
letters in this order is that wherever a strip is put into the
machine there 7t± is the same arrangement of letters on either side
of it. If this were not so it would be necessary to have one
•rod square’ for the wheel when in the R.H. oosition and anoth r
for the other positions.
Chapter II . Elementary use oof rods
Th e rod square and inverse rod square
It Is convenient to have a table giving immediately the effect
of a wheel in any position. We oen make this out in ’the form of
a square measuring 26x26 small squares, the oolumns being labelled
with the numbers 1,...,26, end the rows labelled with the letters
of the diagonal, say qwertzu.... If we wpnt to know the outnut
letter whioh is oonneoted to a given rod nolnt we look in the
row named after the rod point and the column named after the
rod position fpf the wheel. Thus in oolumn 18 and row e of the
F VV m
purple square/ we find's, and looking on th e fixed oomic strips
(Fig ll ) where the purple wheel is in rod position 18 we find the
rod point E connected to output point R
- i -1 i
v/ ro X.
a
w
Ps
C
K 7? \/4fR
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w
W X
c
k ** V-
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w
triune
This square is known as tl\_e’rod square 1 for thja wheel; its rows
are known a s ’ rods r and its o olumns a s ’ uprights * .
We oan make out a rather similar squa©£ in which the rows
are teamed after the output letter* and the letters in the
squares are th e rod points. This is oalled the inverse square.
It should be notioed thet in both squares as one proceeds
diagonally from top to bottom and from right to left the letters are
in the order of the diagonal. Hence the unm name. That this must
backwards
happen is obviousfrom the fact that if one prooeeds steadily roun d
the E,W, as the wheel moves faraard one will always be in oontact
with the same point of the R.H.W. end therefore oonneoted to the
s n me point on the left hand side of the R.H.W. This point is
moving steadily round and therefore the rod points describing
its position move backward along th e diagonal.
cLsru-
Encoding hn tfife rods
Fofc th e purpose of deooding without a maohine, and in oonneotion
with mahy methods of finding keys it is convenient to have the
1-3 9 *
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rows of the rod square written out on actual oardhoard rods, in
guage with squared paper. Let us suppose that we wish to deoode a
thaxfailcnciHg message beginning
x^zxz QSZVI DMFPN EXACM RWWXU JTUT5T NGWX DZ. . .
of not more then 30 groups, that we know the wheel order to
be III I II (Green, Red, Purple), the Ringstellung to be
26 17 16 13, and the Spruoh sohluessel to be 10 5 X& 1
the window position
i.e. that the xtadaxxxBsittsia machine should be set to 10 5 16 1
A
xxxxi a and the deciphering then begun. We first work out the
turnovers in terms of rdid positions. Wheel II has window T.O.
E-F i.e. 5-6, and sinoe the Ringstellung for this wheel is 13
the rod T.O, is 18-19. The middle wheel window T.O. is Ififc N-0
and the rod T.O. is 24-25. N-xt we transform the Spruchsohluessel
10 5 26 1 into rod values by subtracting the Ringstellung. We
obtain 10 14 10 14 , and we oan now write itacxrHtxHiwliiBHXXHf
dHnxlxSzKx over the letters of the message the rod positions of the
R.H.W. at which they ere to be enciphered, remembering that the
a aa.iki.am window position «t which the first letter is enciphered
is not th e Spruohschlue sel but its successor. We can also mark in
the turnovers. Over eaoh section between turnovers we oan mark the
position of the middle wheel. As the message is aaiy not more than
150 letters no double T.O. will be fce^ohed and the U.K.W. will
be at 10 and the L.H.W. at 14 throughout. We wan work out the
effect of these two wheels for this message onee and for all.
We set up the comie strips for the U.K.W. and L.H.W, to this
position and read off the pairs of M.W. rod points which are
connected through them. (The fixed comic strips Fig II have the
U.K.W. and M.W. set to this position)They ere <\o, ev, ba‘, kg,
/ / y y y Lyy
sx, wc, mj , td, fk, fi, yu, zl, From these we wish to
obtain th e connections between the right hand wheel rod noints
far all relevant positions of the M.W. If we set up the red rods
10 - 11
15 16 1IH18|19 20 21 22 23 24 25 25 1 2 3
szv|idVfpnexacm
4- 1~. b . C
h
ao cording to the pairs qo, ev,... (see Fig 13 ) . In any column of
S*>'“ f
the resulting mail cm will be found the letters of the alphabet in
pairs; these pairs are the R.H.W. rod points whiohrare oonneoted
together through the U.K.W. , L.H.W. and M.W. with the U.K.W. and
L.H.W. in the position 10 14 and the M.W. in the position given
*t the head of the column in question: this can be verified fron
Fig u in the case of column 10. In order to decipher the port tfif
the message before the first turnover we set up the purple rods
according to the pairs in oolumn 10 of Fig <3 . This set of nairs
is called the ’coupling of the R2H.W. rods’ or simply the
’coupling*. The pairs of letters in the various oolumns of the
purple set-up are the possible oonstetions when the U.K.W.^rjtxWx
suxfltx ttgatxyjfanpi L.H.W. , and M.W. h--ve the oositions 10 14 10 °nd the
R.H.W. has the positions given at the head of the dolumn. We can
therefore use ±±rf»r the set up for deooding up to the first T.O.
Afterwards we have to rearrange the rods with the ooupling inthe
11th oolumn of the red rod set-up (FIGS ^ )
Chapter III. Methods for finding the connections of a machine.
Alphabet 3 and boxes
For any position of the wheels of « msohine the letters of the
alphabet can be put into 13 pairs so th-t th e result of enciphering
one member of a pair is the other member. These pairs are usually
written one under the other and oelled ’the alphabet* at the
position in question. Thus the alphabet for the wheel order
Green Red Purple end rod position 10 14 11 17 is
MS
VL
ZU
HY
JE
TR
OG
IF
XD
EC
AQ
BW
NP
The order in which these are written is immaterial.
When we have two alphabets to deal with it is sometimes
helpful to describe both alphabets simultaneously in the form of
8*box’ . Take for instanoe th e two alphabets
* f 1
VM VU
ZJ ON
ES JW
GA HI
NP TM
XR FG
OF EZ
HI LR
LB Q3
DW XP
YT YK
UK AC
QC SD
To foim a box from these we choose a letter at random, say T, and
±«®jDcCBrx±fcx±K write it down with its partner in the first
alphabet, Y, following it, thus TY; we then look fd>r f Y in the
seoond alph abet and find it in YK; we write the K diagonally
downwards to th^e left from Y, thus TY ; now we look for K in
K
>7
1
the first and finding it in KU write TY . From this we get to
KU
TY and TY , but now if we were to oontinue the nrooess we should
KU KU
V VM
get
TY
KU
VM
TY
KU
VM
TY
We therefore draw a line, seleot a new letter, lisay, and
start again, writing our results below what we have already
written. Thus we get
TY
KU
VM
W
m
OF
GA
CQ
BL
Eventually when
completed ’box*
Ty
KU
VM
W
FN
OF
GA
CQ
BL
Si
ZJ
WD
SI
there are no letters left we stop with the
(*A box)
There ere various remarks to be made about boxes. A box
completely determines the alphabets from which it was made. Also
it oan be written in various forms depending on the choioes of letter
whioh are made during the prooess, but two different boxes made form
the same alphabets oan always be transformed intopne another
by a combination fag/ the processes
i) Rearranging the order of the compartments
ii0 Moving a number of lines from the ton of the oomnartment
to the bottom, the order of the lines remaining the same
iii) Rotating a compertment through 180° about its oentre, and
<7 ' 4 ' o
then rotating each letter^ through 180 about its centre
At first sight it would seem possible that tn making a box
one might reach a state of affairs like this AB
CD
E.
and thet EA oocurs in the first alphabet, and one would not then
know wh°t to do. This is not actually possible as EA in th e first
alphabet would oontradict AB, For the same reason it is notpossible
to have E coupled with any other letter which ehs already oocured.
If we think of the oolumns in a compertment of a bo* we see thet
the effect of going down the left hand column of a ho compartment,
or up the right hand column gives the result of enciphering aletter
with the first alph abet and then enciphering the result with the
second. Consequently if KsxaxKxflxKKxtk* instead of being given the
alphabets we have the result of this double encipherment we shall
almost h^ave the box. We shall not know how much to slide the opposite
Aides of a compartment relative to one another, and in the oase of
oompartments of equal size we shall not know how to pair off the
sides.
The effeot of enciphering first with then with fi I shall oall
’the permutation likewise the effect of enciphering with
then then ^ will be o^lled J . For these permutations
there is a notstioi^imilar to the boxea. However this kind of
'general box' does not enable one to recove tt the original
alphabets. It is also more convenient to write them horizantally
(the seme applies to ordinary boxes, but thje tradition there is
firmly established). As an example of the notation
p. * (%KLAIYSUHFP ) ( TCWMZB ) ( DEXVRN ) ( J ) ( 0 ) ( Q )
This means that G enciphered at (giving A), and then at
(giving C) end then at gives K, likewise K enciphered x
with gives L, P enciphered gives G, end J enciphered gives J.
With the same notation the alphabet yL oould be expressed in the
form (VM) (ZJ) (ES) (GA) (NP) (XR) (OF) (HI) (LB) (DW) (YT) (UK) (QC().
If the letters of a pa it of alphabets ere subjected to a
substitution, and a new box is made up from the resulting
alphabets the sizes of th e compartments of thds box will be the
same as in the original box: in fact thds box oan be obtained
from the first box b 7 subjecting it to the same substitution,
(exce pt possibly for order of oorapartments eto.): e.g. if we
subject the alphabets to the substitution
ABC DEFGHIJKLMN OPQRSTUVWXYZ
ZDGYTNBHF IKOLUEMSRQCJAVXWP
( Z to repl8oe A eto.) then we
n'
AL
/*'
AJ and the box
%
PI
EU
KJ
TQ,
IV „
NB ^
AL
VI
MU
m
TP
EN
HF
OR
BZ
OD
SD
GS
YV
xm
DO
WC
WK
JK
ZG
PI
SG
QY
VY
UM
B
get the alphabets
Conve -sely if we are given two pairs of alphabets XjU and 0"
such that the sizes of the ooranartments in the box ere the
same r -s in the f (T" box, then it is ->os^ible to find a substitution
whioh will transform \ into £ and into<T~ (in faot usually a
great majqr such substitutions). We have only to write the boxes
in decreasing oompartment size(say), and then a substitution
with the reqjured property will be the one whioh transforms
letters in corresponding positions into one another.
The sizes of th e corape rtments in a. box, end the lengths cf
the kjc xxixx braoke&s (cycles) are important, as they remain the
same if all th e letters involved are subjected to the same
substitution, (which might be a Steokering) . ftaxxfclra: If we write
down the lengths of the cycles of a substitutions in decreasing
order we obtain what we odl the 'doss' or the ’shape' of th
substitution, e.g. the class of p 'bove is 11,6,6,1,1,1; with
boxes there ate two ways of describing the shone, eith er by the
lengths of the compartments or by the numbers of letters in them,
fct It is always obvious enomgh which is being usedj^The following
information about frequencies of box shapes may be of interest.
26
25$
24,2
13$
22,4
7.3$
20,6
5.4$
18,8
4.5$
16,10
4.0$
14,12
3.9$
22,2,2
3.7$
H7T
The phenomena involved
Before trying to explain the actual methods used in finding
the connections of a maohine it will be s well to shew the
kind of phenomena on which the solution depends.
The most important of the Phenomena is this. Suppose we are
given the alphabets at the nos it ions X KKfl .3 fr) Q fXA .X REA FKA WMA
and also at REB FKB WMB then there is ^substitution* which
will transform the alphabet REA into REB, FKA into FKB etc. The
letters of
substitution is that which transforms^ he column of the rod
square corresponding to nosition A into the letters on the same rod^
in column B. When we ere given complete alphabets we can box RBA
with FKA and REB with FKB, and the substitution will have to be one
which transforms the first box into th e seoond . As an examine of t
thin ohe nomen
on we
may
ta T® the
alphabets
and boxes
REA
REB
REA
REB
REA
REB
FKA
FKB
WMA
WMB
FKA
FKB
WMA
WMB
EX
RO
KH
ZJ
TW
XI <
EX'
. RG
EX
RO
UL
FU
JQ
NP
QD
PG
UL.
FU
TTE
TU
HG
JM
NL
EU
ZF
HB
NK
EZ
KN
ZE
CD
AG
GC
MA
RN
VE
HG
JM
RT
VX
YV
KL
ZR
HV
VJ
LN
CD
AG
WI
ID
FS
BY
10
DC
OC
CA
MQ
SP
PB
TW
RT
VX
PA
TO,
KL
ZU
JZ
NH
YV
KL
QM
PS
BW
WI
GS
MY
RT
VX
JZ
NH
WI
ID
TV
XL
BY
WK
VY
LK
FS
BY
BP
WT
SY
YK
IP
DT
SF
YB
Gfl
MJ
AO
Q.C
MD
SG
HM
JS
WI
TE
m^
SP
JZ
NH
EF
RB
AU
OF
OA
CQ
DC
GA
NK
EZ
UX
F0
XE
OR
PB
TW
OA
C£
The substitution whioh will transform REA inti REB, FKA into FKB,
REA REB REA REB
WMA Into WMB, the box FKA into FKB and WMA into WMB is
ABCDEFGHIJKLMNOPQRSTUVWXYZ
4WAGRBMJDNZUSECTPVYXFLI0KH
In this example the alphabets Irve been written out in suoh a way
that ifc® a letter and the result of an lying the substitution
occupy corresponding positions. Of oourse if our alphabets were
data from whioh the substituion was to be found this would not
generally be the osse. Our problem would be to arrange them is
y a m fr i rT or the boxes made from them, in suoh an order •
We might for instance be given the alphabets in the more or less
alphabetical order
tioin. H&b EKA EKB WMa.
WMB
REA.
SKA
REB REA REB
: EKB WMA WMB
AO
AG
AP
AM
AU
AG v\
AO
AG
AO'
AG
BP
BY
BW
BR
BY
JfcH
IW
SP
CD
PS
CD
CQ
CG
CD
CO
DT
BP
NH
QM
JM
EX
DI
DM
EU
DQ
EV'
CD
VX
HG
YB
ES
EZ
EE
FO
EX
EQ
m
LK
SF
HN
GH
FU
HK
GS
EZ
GE.
JZ
YB
ZJ
LK
IW
HN
10
HV
GS
afe
RT
RO
VY
WT
JZ
JM
JQ
IW
HM
DT
VY
E0
BP
DI
m
KL
LN
JZ
IP
J S
SE
EZ
IW
XV
LU
OR
RZ
KY
JV
KW
EX
JM
TR
EZ
MQ
PS
SY
LX
KL
LN
UL
NK
UE
RT
TW
TV
NP
NR
MY
NK
TW
LU
I
VY
VX
UX QT TW
Jl
HG
m
w
and
thjan mak«
from^th^p
boxes
on t
he right. From the right hand
pair of boxes we se' that Emust become either 0 or R in the
substitution, and we oan try both* hypotheses out ±n arranging
boxes
the first two aMnchBtx correspondingly. If the first box is left
as it is, the oo "responding rearrangements of the second are
W AG
.PS SP,
' iMJ' VX
ZE LK
UP YB
OR RO
BY FU
KL EZ
XV JM
The first of these rearrangements is impossible. It implies for
instance that In the substitution C becomes H and M beoomes P
but
sdaarxsKx in the third box C end M occur on opposite sides of a
M*.,i P
compartment while in the fourth t tor pre on the s n rae side,
six
Aotually we have in the^alphabets rather an embarres de rlohesse .
It would really be easier to work with say the first five
Kj
alphabets and ±ha two con3tatations , AC a* d BH say of the
remaining one. Since B and H oocur three apart in the same column
REB ,
of EKB the pair ofletters of WMA from which BH arises by the
substitution must eocur three apart in one of the columns of
REA
the large compartment of EKA . The only possibility is thjat BH
arises from FZ, and we o e n check th e result with the A0.
I
qf
We make use of a third phenomenon when we have found some narts ml
of rods. Suppose we find the substitution which transforms
the fix^st column of the purple rods into the third
G EC A
Y7I I
T CD E
NIAD
B ST R
H ZG C
F HZ W
IUBN
K NR X
0 TL Z
L QVM
UBIV
E OQ J
M WN 8
SMPT
and the substitution which transforms the third column into the
*
fourth is
(,
( JYBNSLZWPTRXIVMQ ) ( CADEOG ) (HXJ) (FK)
These two substitutions are of the same ’ shaue8, and if we write them
like this
f YVLQGEOTC IUBSMWR ) (NFHZDK ) (PA) (JX)
( JYBNSLZWPTRXIVMQ ) ( CADEOG ) ( HU ) ( FK )
eaoh letter in the lower line is below the letter whioh is three
places further on along the (QWERTZU) diagonal. We can soe that
thSjisjmust hapoen because if we replace the letters of the
first and third columns of the rod square by those which are
three places along the diagonal and then move the
tesult three plaoes to the right and tl\jree unwards we get the
It is
( ZDKNFH ) ( GEOTC IDBSMWEYKLQ, ) ( JX ) (AP )
fourth and sixth columns
A rather similar Phenomenon is useful when we in know the
diagonal of the maohine. In such a c e se we c n n m-ke o correction
to out oons tat ions transforming them into oonneotions between
the oontactsxafcxtiiacxS on the right of theR.H.W. instead of between
oontaots of the Eintrittswalz. The oonstatations when so
transformed are described as 'added up' or 'buttoned up* . The
process can be carried out withtwo strips of oardboard with the
diagonal written on them, and in 6ne case repeated. As am example
to make quite ole~r what this adding up prooes • is take the
fixed ooinio strips Fig 11. The alphabet for this position of the
machine is (CB) (FR) (TV) (XO) (JK) (WQ) (AG) (FY) (BZ) (HM) (IL) (EM) (US)
The ad led up alphabet c*n be obtained either by tracing through
the wheels from the purple column on the right b«ck to this
column again, or by applying the substitution
QWERTZU IOASDFGHJKPYXCVBNML
YXCVBNMLQWERTZUIOA'SDFGHJKP
to the ordinary alphabet. It is
(FR) (TV) (BG) (DQ) (10) (XY) (WZ) (AS) (HN) (UK) (LP) (CJ) (ME)
Instead of tracing the oufcrent through from the right hand purple
oolumn in Fig 11 we can of oourse treoe it through from the left
hand purple column back to this oolumn again.
This gives us a very simple picture of how the added up alphabets
between turnovers are related; one is obtained from another
simply by a slide on this left hand purple oolumn, i.e. a slide
on the um last upright of the rod square. For instance
if on the £±xad comic strips Fig 11 we move the R.H.W. to rod
position 15 we have the added up alphabet
(EA) (RD) (VM) (10) (FN) (UB) (LF) (GW) (YX) (CT) (QJ) (KZ) (HS) (
whioh daii be obtained from the added up alphabet at rod position
18 by the substitution
The saga
Suppose thet one w p s left alone with an enigma for half an
hour, the lid being looked down end the Umkehrwalz not moveblw,
what data would it be best to t r ke down, end how would one use
the data afterwards in order to find out the connections of the
machine ? Can one in this way find out all about the connections ?
This problem i3 unfortunately one which one oannot often apply,
but it helps to illustrate other more practical methods.
It is best to oocupy most of one’s half hour in taking down
complete alphabets. At least nine of these °re necessary, as
d8ta the number of possible different data must be at least
equal to th e number of possible different ix solutions. Now the
n umber of possible different diagonals is xpxxarrt 261, the
number of ways in which one can wire up a wheel is also 26$, and
the number of ways in which one can wire an Umkehrwalz is
approximately (261)^ , so that the number of possible solutions
is about (261 ) 9 ^ 2 . The number of possible variations of an
alphabet is about (261)% so that th e number of possible
, . g/2
variations of nine alphabets is about (261) which is the
number of solutions.
The praotioal minimum amount of data is surprisingly elose
to this theoretically minimum. It is possible to find the
connections with 9 properly chosen alphabets and 10 other
oonstations properly chosen. However in order to shorten the
work I shall take an example where we are given 11 alphabets
andIO oonstatations.
2 4 '
Data for sage
v a / j \ s y
l AAA ✓ AAC / ABA ! ✓ ABC y CAA CAD
'■ AAB AAD ABB ACA BAA
/
ADA
CAC
xxxanronmr
ACB
DAA
AL AD
AI
AM
AK
AE
AW
AM
AS
AQ, AZ
SO UQ,
MJ
HX
BS BG
BY
BS
BO
BS
BV
BP
BO
BV BN
ZJ
LB
IL
VS
tf a
CE EK
CT
CH
CF
CR
CZ
CE
CP
CH CO
DH FV
DM
DR
DE
D q,
DX
DW
DJ
DU DF
FM GZ
EV
EO
GQ
FL
EJ
FG
EU
EP El
GR HN
FN
Fq
HW
GV
FO
HL
FQ
Flrf^GL
IK IT
GX
GP
IX
HK
GU
IZ
GV
GM'HX
TZ*
JN JY
W
IJ
JP
IN
HI
JO
HY
IZ JR
OZ LU
JO
KX
LS
JP
KR
KQ
IL
JO KP
FV OQ
KZ
LT
MY
MO
LQ
NU
KT
KN MY
QW PS
LW
UZ
NR
UY
MT
RS
MX
RW QV
TY RX
PQ
VY
TZ
wz
NS
TX
NR
ST ST
TJX MW
RS
NW
irv
TX
PY
VY
WZ
XY UW
V
There will he a substitution which transforms AAA into AAB,
for f indin g such p substitution
ABA into ABB and AGA into ACB, Following the method/exola ined
AAA AAR AAC
in th e last paragraph we form the boxes ABA, ABB and also ABC
which will be needed l°ter
AAA AAB AAC
ABA ABB ABC
GR BC OJ
NJ RX EV
FV TI BY
TJX NH PQ
IK Hi LW
X/
CAA
CAC
A S
HX
BO
VS
6P
DJ
EU
FQ
GV
HY
IL
KT
MX
NR
WZ
AAB
We want to rearrange the box AB& in the w~y that was done at
A AA
the bottom of o. .The substitution which transforms ABA
AAB
into ABB must also transform two oonstations of AGA into SO
and ZJ. The only constatations fxn of ACA from which SO could
have arisen ere LH, VY. If OS arises from gjLUagx LH we
should h^ve to have a substitution which involves ZJ arising
from OE in ACA, and this does not exist,
A TiiriV" ..v, . -,-t n n mg . However If we rearrange it so
that OS arises from VY we find ZJ arising from cnn
AAC
similarly arrange ABC to fit with them and agree with CAA and CAC,
‘ A AA AAD
and fit CAA to fit onto CAD agreeing with BAA and BAD.
Rearranged
AAA AAB AAC
AAA
AAD
Rearranged
AAA AAD
A BA
. ABB
ABC
m
CAD
CAA
CAD
Al
vf
GX
AL
UZ
AL
TL
SB
LU
DM
IK
AM
IK
GP
OZ
YJ
TC
TY
YV
TY
KX
TY
BS
ZK
HD
QF
HD
HC
MF
BC
RS
JN
DR
JN
OE
CE
fiX
NF
RG
TI
RG
IJ
3JH
TI
OJ
VP
EO
VP
RD
WQ
NH
EV
CE
CH
CE
FQ
GR
KE
BY
UX
XK
UX
VY
NJ.
AD
PQ
MF
PG
MF
MA
PV
QO
LW
QW
LT
QW
ZU
UX
m
AI
zo
SB
ZO
WN
IK
ZG
HU
BS
NW
BS
BS
We oan noowtarite down the Darts of the rods whioh are in the
oolumns corresponding to the window positions A? B,C,D though we
do not know the oorrect order. They are
AVGT
YSKX
WNEU
UMAV
LFXL
MBRM
QHVZ
XWIY
SLD3
FCSA
GKBJ
IZHG
BUMB
CRNF
REYI
KGUP
OYTN
EXFQ,
NAPE
JDQO
DTOC
HJLD
HIJH
VOWR
The substitution whioh transforms the letters in the first oolumn
o
of these rods into those on the same rods in the second oolumn is
(AVOYSLFCREXWN) (BUM) (ZJDTPQHI) (GK)
That whioh transforms the seoond into the third is
(VGUMAPZH) (FX) (LDQ) (YTOWIJCSKBHNE )
and tint whioh transforms the third into the fourth
(GTNFQQCWSMBJH) (XLDSAVZK}.(EUP) (YI)
These three substitutions have now to be arranged one under the
other in suoh a way that the substitution whioh transforms the
third into th e second is the s-^me as that whioh transforms the
seoond into the first, this substitution being a slide of one on
(?*) Stca^i
the diagonal. Clearly HH-) in the t -k lrd has to fit unde 7 ' either
«,*. <r&
(M) or (XF) in the acaramot first; if F is under G we cannot fit the
second end third together, for F oocurs in a breoket of 13 in the
third, and G in a bracket of 8 in the secAnd. if F is under K we
can fit the three together like this
(AVOYSLFCREXWN) (BUM) (ZJDTPQHI) (GK)
SKBRNEYTOWUC (QLD ) (VGUMAPZH) (XF)
(NFQOCWRMB JHGT ) ( PEU ) ( KXLDSAVZ ) (IY)
The diagonal is
APQBORYFKYZHIXGJWELUDlvOTCNS
xn^xxax«HnxxiKkRxsrKt Of oourse we do not know where the diagonal
’starts*, but with a hatted diagonal like this it does not matter.
We can use the diagonal to put the' rods in order and to give them
n^fmes. There is likely to be an error in our naming, because
we shall not know where to start naming t tamer.
* order to write the letters in the
strip ftfr theyheel, but di6 not
known/
6ioh tru" /Window positions and
rUUfLUIlg are ir?volVea
either th_h rows or th^e columns , Th^jB difficulty about naming
th^P columns simply means that we do not know the Ringstellung
or th*_e absolute positions involved. If we have the oolurans
correctly named but the rows wrAngly we shall have the wheel
right except that the plate oontacts are rotated with reject to
the spring oonteots, I& is very difficult to eradicate thisj is*
*x8rotxmswtx*k*s
It can only be done If we have 8 great de^l of information
about aotual window positions end Ringstellung, e,g, if there
is a Herivelismus or if the letters bf the Ringstellung are
restricted to be allldifferent and notwo oonseoutive in the
alphabet exoept Z and A,
Our set of rods is
tS
EHG- z
HIJH h
XWIY i
EXFQ x
GKBJ a
VOWR j
RTOC w
LXjFL e
KGUP i
JDQ.0 u
MBRM d
OYTN m
FC8A t
NAPE o
PQJD n
BUMB s
DTOC a
CRNF p
YSKX q
AVGT b
ZJCW o
WNEU r
SLDS y
UMAV f
TPZK k
QHVZ v
and we can n^ow transform all our data about other alphabets
into th_e form of data about rod oouplings. The ones we need first
are
AA
ah
be
ou
dt
fi
gw
Jn
kq
lz
mo
P*
rv
st
AB AC
ax yw
bl eh
ow bd
dtypx
ejr gt
f 5 ki
ko zo
ms vl
nu ns
pt um
gv jq
rh fc
iz ar
AD
fv
es
From th^pse we can get the upright of the middle wheel. The
first step is of course to add up the alphabets. Here they are
added up with Z as standard
AA*AB*AC *AD*
pi qj vu hi
ol rd dg mb
nd si ;£ c
me to ow
kx uk es
Je ve jh
ws zy xf
vb op im
uh am pq
tr bn tn
qg wh lr
y z fx za
ef gi bk
AB*
A C* so Jcind: as to find the substitution whioh transforms
AA* AB*
A B* into AC* and AC* into AD*
AA*
AB*
AB*
AB*
A C*
AC*
rearranged
Pi
aj
#1
m
hw
SC*
croc
po
Je
ot
St
xf
yz
ME?
nb
Bk
aQ
n d
ku
xfc
nh
ig
rt
ve
K*
®s
dr
ol
si
is
XB
Is
sw
rd
*r
fcx
ev
hu
gi
is
Sx
uk
&
me
zy
fee
ft*
tar
s
i
3
g
S
$
This substitution sends each
letter of the upright of the middle wheel into the next on the
upright; h^ence the uptight is
±2 lsezftrdgpjyxniqohukbmwvao
As we added up to position Z as standard this upright is(the
‘ upright for position Z. We oan make out part of the rod square
there being
from it ^difficulties about where to begin as before
ZABCD
LNJHB z
SWKOL h
EVRUP i
•SYDQW x
FMBEZ g
TOLHC 1
RUINH w
DXSIQ e
GAXBV 1
PGOZD u
JRHMK *
YITVN m
XCZSU t
NHADF C
IPMKT n
Q.TVWO s
CZERS a
HLYAE p
UTTPLI q
KQ.HXR b
BDGYT o
MJFCA r
WKNPG 7
VSQJM f
ABWTX k
OECGY V
We can now transform our remaining date into informs tiori about
couplings of the middle wheel rods. By sliding the diagonal
up the side of th e rod square we oan get the couplings
immediately into added up form
A* B* B*
A*
B*
c*
D*
B*
C*
X* rearranged
c
re
as
yay
kd
ra
as
wl
bt
bn
bli
OX
si
gz
or
oe
or
ol
Wj
eq
do
di
do
dr
kg
Jk
vf
fo
eq
ez
zv
tx
nb
gk
tv
fn
fo
ph
iu
& y
gz
gs
di
wl
my
jw
hp
hw
un
or
as
Is
iu
xp
bt
do
gz
mx
Jk
Jq
xm
vf
eq
n u
lw
kt
yk
nb
Jk
P q
my
mu
pq
iu
tx
vz
tx
VO
ec
my
The
left hand wheel upright is
rwdmqxeptznschkvbgfiyjoual*
zhixg jweludmtcnsapqboryfkv
and under it has been written the diagonal. This serves to transform
A or A* into the UmehrwalB connections. They are
yv,fs ,oe ,zw,oi,mu,rj ,qx,pk,nd,ht ,bg,al 1
2!
’ Adding up 1 method
Most practical methods of finding the connections of the
maohine depend oil getting a lox^ orib , either by ’reading on
depth* (see Colonel Tiltman's paper )or by
pinohing. In many cases we expect the diagonal to have some
special value, (e.g. qwertzu beoause tbs original commercial
machine had such a diagonal). In this case th e amount of orib
necessary is not very much . To estimate the amount of material
that we have it is best to work out
(Length - Sl5)X square of average ^corrected depth’
Calltggxthis the ’material measure’. By corrected depth we mean the
exrx aotual number of o onsta tat ions , so that this oan never
exoe^ed 13. As regards the amount of material necessary, it will
almost always be impossible to get the wheel out with leqj^ than
a measure of 90, from $0 to 140 it will be a matter of ohafaoe whether
it oomes out or not. From MO onwards it will always oorae out, but
with increasing easeps the material measure mounts up. With a
material measure of ^00 it is so easy that the trouble of adding up
further materiel would be more than would be gained in shortening the
further work. The method is anracxiarai' essentially the same aa we
used for finding the middle wheel in the case of the sega. Here
howevwr we have to do with oertiel alphabets or even single
coQ^statations instead of complete alphabets. We cannot therefore
do any boxing. After we have added the material up we take some
hypothesis about the upright, e.g, that F immediately follows K
and work out its consequences. If for instanoe we find the
padded up. I shall rat omit to mention this in future) oonstatations
® and ¥ immediately following one another we oan infer that
T immediately follows R on the upright. This we may express in the
form
|< P - 'K ,
the dash denoting logical equivalence. We folio™ out the consequences
until we reach a confirmation or a contradiction. When there is
/■K- HI' »'-***- 1 /* / k ,
Hj. 1
HI 6 * wrU/ \i rF
b~*> /W'
i
32 .
plenty of materiel we do not usually start to work a hypothesis
unless there is going tcfbe an Immediate oonf irmation, e.g. If
TC Implies HI from two different parts of the crib. This will
mean to sey that the oonstatations I and 0 oocur torfra oonseoutively
twice oyer. Alternatively we can say that 8 oocurs twioe over
at a oertain distsnoe , and that C also nocurs twioe over at the
same dlstenoe. In order therefore to find these profitable
hypotheses we have only to look for repetitions of oonstatations
(half-bombes as they are rather absurdly oalled) . For this reason
and rattraintk also because later we will want to be able to spot
oocurrenoes of a given letter at a glanoe, we nut oar materia 1 as
we add it up intothe form in Fig ,
How to take a particular problem. We are given material sir deep
and 100 long, end we eipeot that the diagonal is qwertzu. Our
material is
MYC..
NGJ. .
RCA. .
YlD..
DAS..
TTV. .
YON..
RMI. .
OFL..
VQO..
MOX. .
NJQ. ..
Ojmust apologise for it not making sense) . » « . .ii* l i ra rmfcqnit
ifitx.
We deoide to try out the hypothesis that there is no T.O. in the first,
seven columns, and therefore we add up the oolunms 1-7,27-33,53-59.
getting
Lot..
MJY. .
TBF. .
XAH. .
FUG..
ZUM. .
However we put the materiel direotly into the form of Fig t ^ .
We see numerous hslf- bombes and do not need to make any
analysis of their lengthd in order to find a profitable
Q F
start. The half bombes S and H suggest the two possible
sterts Q F = SH end Q, H=SF (the two strokes meaning e double
implication, not equality 1 ,). The oonsequenoes of the seoond of these
are shewn in Fig 10 . Aoontr^diotion is quiokly reached. The
consequences of QF in Fig LI . The loop QF-ZO-MB-UJ-QF gives
a seoond confirmation, and our hypothesis is now ° virtual
certainty. We now abandon the tree figure for an alphabet with
oonsecutives written against them (FIG 22.). All .roes smoothly
except that there is clearly r n error in our data ps w§ have a
few contradictions. We sort out the good from the bad by using
2 ?
pairs of letters two apart on the upright. Thus JO - AF confirming
fXS JZ,Z0,AQ,QF. When we have oheoked them all we can write out
the upright of the R.H.W,
AQFPEVKYNCUJZODXMBSHTIRGWL
We then have to find the upright of the M.W, To do this we use the
same 'rooess as we did with the saga. We have to find the
added up couplings of the middle wheal. This oan “ctually be
done without either adding u-> separately or writing out the
two
rod square, simply by having Jmovable strips with the upright
and qwertzu written out on each, and sliding these above the(added u]
cribjtill the constats tions agree with pairs of letters on the
strips direotly above. We then read off th^p coupling from the
row of qwertzu letters, taking the pair of letters in oolumn
1 for columns 1-7 of the crib oolumn 2 for 27-33 ete. Under
one
Fig ie [ is shewn the strips as xfesx set for reading off asms of
the added up couplings for 53-59, viz aq . The added up
couplings that we feet are
1-7
qp
?x"33
53-59
08
79-85-
wb
qs
wj
XV
ef
wu
eg
tr
ry
ek
th
ffc
tn
rn
rv
ql
zu
to
zx
up
ix
zy
um
oy
os
ia
io
ds
ag
ov
sk
wb
dm
aj
db
oi
hV
fm
fy
gz
jo
gU
pn
em
kl
Pi
cl
ka
_ hi
>of-*u J
zm
ti
(jr~^ fl— ^
Boxing these together we get
1-7
27-33
53-59
27-33
53-59
79-85
qp
hx
qe
Ik
zy
ks
ef
fm
db
md
uw
w3
3d
np
tn
”bg
urn
ry
ek
eg
zu
sq
zx
wb
ai
vr
ga
ov
th
ix
rn
Zf
hv
Pi
oi
os
£t
21
When we fit these boxes together we fail miserably, and so we
have to assume that there is a double T.O. somewhere in spite of
the boxes all turning out the same shape. We find th«t this
is between the first and second alphabets, and that the
r mainder can be fitted together with tl^fi upright
wbnho ovrt ixlyazqgpf kmseud j
I will ive r second example of the 'eddlng up' method for a
ease where it is only just possible to get the problem out.
The materiel is given in Fig ell ready added up. There ere
no 'equidistenoes' (half-bombes with equal distonoes) end so we have 1
to make an analysis shewing all the oonsequences of any hypothesis
that one letter follows another on the upright Uig . Fot
instance from the enalysis we dee that AV,HT,NF,£A, are
all consequences of Hi. The penoil letters round the outside
ittw were put in tojhelp with the making of the analysis "Hd were
used in connection with oolumn^s 32,33 of the materiel. Of oourse
some of the conseqnenoes will be felse owin_r, tojturnover, but as
we e e dealing only with distances of 1 are oan hone to negleot this
without Krrm. We now pick out tar* squares with e large number of
entries in them and follow out th e further consequences of them,
making trees as before, and hoping to find confirmations. When
w get contradictions we leave the tree for the present but have
Figs IK-to
to remember the T.O. possibility . When we get stuok we oan
sotastlmes oontinue using oon^sequenoes which are of th_e form
th_st|two letters ere at dietanoe 2 on the upright. For this
purpose an analysis of positions at which letters oocur is useful
(FIgl4) . xt In pertiouler we need /this at Fig Jb . Now VW end
WY imply VY 8 end PR and RS imply PS 2 and these imply on e
another from oolumns 19,21. We also get Gl® which sterts off
another treln of oonsequenoes involving smother oonf Irma t ion 4 y •
Eventually we get stuok with the bits of tata uptight
YWY
H.ft PRS
UHJK
F0IL.0
B.E
We might try putting in KA as a hypothesis, Shtaxsmta afterwards
try KB eto.(KA appears at first to give confirmations, but these
ere bogus. Th^p only reliable rule about oonf irmetions is to
Hui Lu^rr. .
30 , ~- if Ttrrr^ a oonst-tation out and then see if it
can
he inferred from the hypothesis). We might also try
putting in as many new oonste tat ions e g possible which are
oonsequenoes of those e have and out available information
about the upright, and then start off afresh with some new distance
on th e upright, say 5, But there is a quigker road to success,
H G
Note the constation J in 1 and I in 17, Since we have J following H
and I following G on the uprigh t it seems highly probable that
we have HG 1 ** and JI 1 ^ , If this is so we have this as part of the
u Tight
PGIL*O....UHJK
Hence OB^ whioh implies PK 6 giving us this as upright
FGILNOQPHSUHJK
Prom this we get many confirmations and are able to fill in the
whole of the uprigh t (except Xv/hioh goes in the one remaining plsoe),
t.b-t the T.O. which actually ocours between 24and 25 has not
troubled us at all |
4 U
* -
~~ l—
> G-
'ji <-»
pO 1
I "
I
c^- © —&
31
&)
| a r^'V‘»
^ U
/ £5
r-vcT \
/ . < 3 >-
©
^ nP ''
(wj) — (©H^B ) " &-(&-<¥$> •
y
f-q
1 o
r-^1 ^
wy -TK Si |
fi-S VvV uh
II K
Clicks et twenty-3 ix^d is ta nee
This is 8 method for finding the connections when we do not
know the diagonal. It is very similar to the xacgxpdt beginning of
the saga, in prinoiple. It depends on making hypotheses nxx±*
about pairs of letters being on the sax© rod, and drawing conclusions
xmf thKxxxniacxktx to the effect that other pairs of letters are on the
same rod. Suppose for example that in our crib were the following
constate ions
5 6 31 32 57 58 83 84
AE FE TU PU
FG TR PR AG
We migh t make th e hypothesis that on the rod whidh has A in
column 5 there is G in column 6, We could then infer that, there
was another rod with Fand S in oolumns 5,6, and likewise rods
TR, PU and this confirms our hypothesis that there was a rod AG.
Proceeding in this way we can with sufficient material find
sufficiently much of some of the rods to be able to find the
die onal by the sa a method. The amount of material needed is
very great. We adopt a measure similar to the one for 'adding up'
viz
(length-39)^ square of average corrected depth
I believe it is practically impossible to solve any problem with
this measure less than 2000. It XBskscxsclmBxicxkH should be possible
for 3000 but might sometimes involve *> gre-t de«l of leboufr. With
the example given here tbe measure is lf.ti.eo .
When th^e materiel is sufficient w* avoid taking h_ypotheses at
random, and choose ones wh Aoh we can see ±mmi4±act®kyx±B without
very much analysis, to lead to an confirmation. This would be
the case for example with these oonstations
56 31 32
S E RE
V D V D
Either the hypothesis that E follows R or thet D follows it on a
rod would be immediately confirmed. In the absence of other
information the probability thpt one or other of these
hypotheses is cofcreot is about 79%. Our first job therefore is
to look for suoh configurations of letters. All that we have to
do is to analyse the arsctariatl constatations whioh have acx$±xra
the same right hand wheel position, and ring round any repetitions.
We then write out the ringed constatations on a separate sheet
(Fig 2/f ), With the first ooourrenoe of each const^tation we
give a number shewing how far on the other occurrence is.
This nhacrfcxaci plan also shews us where the T.O. is likely to be.
It should be mentioned that in the oase of this material there
known to be 13 apart
we’ e two turnovers. The prinoiole of shotting the turnover is this .
Consider for example the constatations HE at b,II and b,X and
JE at i,IIand i,X, The first pair of these constatations
shows that there must l\_^ave been a BDupiia pa it in oommon between
the coupling at b,II ano b,X , Likewise there must be one in
common between those at J.,11 and i,X • It is therefore fairly
likely that there is no turnover between b,II and fcpE i,II f as if
there had been it would have been quite likely that after tbs
T.O. there would no longer have been a pair in oommon in the
couplings. The- evidence from 8 single such instance is rather
slight, but with afWoh material as we have in out present problem
we oan i ix it with no doubt at all, as ooourring
between z and a and between m and n.
It is worth while writing down ell the favourable hypotheses
Yrnder the pairs of columns of the rod square involved fFig 3 ^ ).
We have done this only for the part a to r&>, and find that in five
oases there are two favourable hypotheses viz. col. b with e
ool. bwith h, col. d with j , col, e with i, and ool g with j. We
hope that in some of these oases the favourable hypotheses will
imply one another, making them both virtually oertain. X5££i£XX£X
of these hypo theses)* re shewn in Figs . The hotation is this.
An expression like OF under the head ’dtnto y means that the rod
with 0 in col. d has 8 F in col. j, and the strokes jiining these
mean that one can be deduced from the other. I” the oase of g into ifl
the two hypotheses a e essentially the s n me and we have an Immediate
confirmation. With b 4>ntb h we find that both of the first
alternatives of the dine hyofcthesis oontredict both alternative^
of the other. With d Anto j we manage to connect the two pacx
hypotheses together and with e «nto i we fail to oonnect but ±tas
one of the hypotheses confirms itself. The information
we have obtained about the rods from this is expressed in the Fig^/a
In order to avoid bogus confirmations in whet follows it is as well
whenever we make a deduction to cross out one of the
constations used int the deduction. ikOT*xx Up to this noint
the crossing out has been done with red strokes slenting un to
the right. (Green vertioal strokes were used to eliminate
repetitions o^t a constetion, red vertical strokes to remove
oontradioted constatations . ) . From now on for a time we will usd
similarly slanting green strokes.
xxxxhxvxxnonrxDckHTrtxfci&xxHdyxQpssrycKxxiiixissxtBcjtaJciyxlsHi
Up to now we have simply been trying to ’get a start’ , and
so long as we could gerjsome faifcly considerable bits of the
rods square fixed we did not very muoh oar© what Darts they
were. But now we have got a fully adequate st«rt, and we should
consider a plan of oampaign. In general what we went is tlmx
most
to have xhx® of the letters of the
rods in columns Gy £ *t* tf',
4 r 4* ^ of which any number may coinoide, provided are
xJWJQxccsEisx xpcix If we then find the oermutetion whioh transforms
col. f into col. ^.expressed in cycles e r - on p /$ or t> 2 -£ ,
and similarly for col. ft** end ool.^^t k. Ai'slide of V' on the
diagonal will transform thdsWinto one another. We get further
information about a slide of K on th^e diagonal by finding
the substitutions that transform col. { ’? into ool, £**♦** t
and ool, lr * ^ into ool. £,*••* u.*K. Between the two sets of
information we should have enough to reconstruct the diagonal
(unless If - 13 and as long as the bits of rod are not too incomplej;
In th^e present oase we oen teke th_ja oolumns c ,d,f ,g, ;J ,k;
giving them the numbers 3,4,6,7,10,11 instead of the letters this
corresponds to 'j 4 * 5 , ey*S , \Jv ^ , v- v H- , r%\ . In order to
get these columns* ’"e look on Fig Ji^for sflytable hynotheses to work
in order to add in the extra oolumns. These hynotl^e^es enable us to
write in extra letters in the Fig /ftaend we continue tojurrite in
letters in tnis figure until we reaoh a confirmation scrtti wh
XBH jsix or a contradiction. Until we reach a confirmation it is
as well to differentiate the letters th^t are certain from the
rest. The hypotheses that we actually used v-ere : uanka o <>hto g
IQ=SE : ginto k 23S=ND. After a considerable amount of work our
ix±x± rods look like Fig 4ft. The lines crossed out are ones that
have been amalgamated with others. We now think we oen start to
look for the diagonal, and therefore make up the nermutations
transforming o into f, d into g , f into j and g into k. The notation
is that of p I ^ , except that we are mostly unable to complete the
brackets, ap^d leave dots,
o into f
. . .DCYQFVJZTAXHIN . . .SGDFR. . ,KE . . .LUB . . ,M. . ,W. . .
d into g
. . .KW'CM. . .ANSY. . .GLIJ. . .TUQ. . . BEBXOR . . . FFZV . . ,H. . .
f into j
. . .QOTK. . .UEJUOR. . .BSZW. . .PEA,. . .CXIM. . .*H). . .E. . .L. . .V. . .
g into k
...XND... (fa) . . .KF. ..TYHZ...MftBLjmjRG...PA.. <-S> _ -0 .. .V. ..
4 g
We have nowto write tl^ c into f permutation over the f± t into <
permutation, end the f into J over thee into k in suoh a way
, i
that ±h* a given letter in acix Sj intorSra f into j stands over
the same letter in ( d into g'and f g into k. To get a start on this
observe the configuration of the ringed letters. This suggests that
we arrange the permutations in this way
DCYQFV JZTAXHIN
25EBXOR
This is further confirmed many times, and we get the permutations
arranged like this
( DCYQFV JZTAXEQN ) MSGOPR
(EBXORMSYGLIJD) KWCMTUQ
(YD) QOTK UHJNGRCX IMPFA
(XE) OTYHZ INDMQBLJWURG
giving us the partial diagonf]^ slide of 1
. . .BCSZ . . .EDNJIHK. . .LXYTOQRF . . .WMGAV . . .UP. . .
Z must he followed either hy Si ifWxiJr E,L,W,or U , If it is
followed hy U we get
( KEMSGOPH ) ( BWUL ) ( KEWLUBMSGOPR)
(HKWCMTUQ) (FPZV$) or like this ( HFPZVKWCMTUQ.7
giving the diagonal slides
(EDNJUK )(...)
(UP)...
both q which are impossible. If Z is followed hy W we have the
hits
MSGOPR KE W LUB
KWCMTUQ, H FPZV
which fit together only as
(KEMSGOER) (LUBW)
(HKWCMTUQ) (VEPZ)
and as before the K configuration makes this impossible. We
oannot have Z followed hy E because of the impossibility of
LUB
EPZV
and the diagonal slide as
( BCSZUPLXYTOQRFEDN J IHKWMGAV )
If Z is followed by L we h*>ve the bits
MSGOPR KE LUB W
KWCMTUQ H FPZV
to fit together, which we find
i cm aVwari , oan only be done like this
fitting KE onto
H
BC8ZUPLXYT0 QRFEDN J IHKWMGAV
After the previous examples thet heve been given It is hardly
neoessery to explain how to get the uprights of the various
wheels after this point. The upright of the right hand wheel
would be obtained by rearranging our bits of rod, snd the middle
wheel by the method desoribed on p. , with luck we might find
other messages on the seme day with different L.H.W. positions
and so find the L.H.W. upright. In the oese thet the Umkehrwalz
is movable this may be rather trioky , but in
suoh a case* there ere or_pbably no gteoker, and we should he
able to solve other days by single wheel prooesses, with the
known wheels in the R.H.W. position, »nd hope for the unknown
wheels to ooour in the M.Y/. position.
In the example given above the diagonal is actually ABOD . . .
with Steoker. We might have tma had e hatted fundamental
diagonal with Steoker, and of course in such e oese we oould
not have said what the fundamental diagonal was. We should then
have had to w±r*x oyxthax prooe^d to try to solve other days
keys by spider methods, without diagonal board, and assuming
temporarily some arbitrary dlagonel^fondsmental diagonal, and
non reoiprooal steokerlng. With two or three suoh keys we
should be able to find the aotuel fundamental diagonal by
comparison of tl^je steokered diagonal.
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Finding nusw wheels. Steoke r knock-out
Sq far we have been dealing with the problem of getting out the
connections of an entirely new machine, or one fcr which we know
no more than th^e diagonal. There is another problem, th_^t of
finding thus connections of some newly introduced wh_eels, th— e
old wheels, or at any rate some of than, refining as well: this
includes ths case of a change of UMkehrwalz.
The most hopeful case for getting out the new wheels is when
one of th^ knov/n wheels occurs in thja R.H.W. position. If the
maohine has no Steoker there is no difficulty. We solve some messages
by singdA whjsel processes. This will be slightly more difficult
than when we know the connections of the middle wheel, as we .shall
three ofc four
h eve to guess what is said in ±:Jps different turnovers. However
iBDcfintxwliactxtac when the R.H.W. rod st«rt has been found from a
guess in one turnover it does not take any titae to test a mot nrebable
throughout the message (the rods on which the various letters of the
message occur oen be written down once for' dll, and the mot probable
punched out and run over the inverse oblong). For slmolicity let us
suppose th_et we have read the mes-agex right thorough. We then h_pve
the couplings in several jnaxmox oonseoutlve positions of the middle
wheel, and can apply the methfcod of p 28, 29 to find its uoright.
In th^ case that the machine has Steoker we nedd rather more
data, and verymueh more patience. The sort of data that one needs is
1 26
a crib of length °bout 70, or else one of length 18 and depth 2, The
trouble about cribs without any depth is that ode uses up the-
a great neny of the constatations arffcgcx between each turnover in
determining the coupling.
An example is shewn of Stoaa a orib of length 18 s^ji depth 2.
greater length which has be n out down to allow for turnover. The
text of th_£ orib is shown at the toe of Fig . XftsxKXKix
xOTSxzlxhxlfxkmxxBXxlxxljE?: We are taking the worst oase of 13
Steoker. There are several helf-bombes in the orib, and we deoide to
work with TW. We have to make ffipcfcx 17576 different hypotheses,
(app) corresponding to the 26 possible different places on the R.H.V.
ft /v r li ft * -S PeRTcH-t'Pi t'tf
F & N ‘I P X r W I OWT)U 3 >LM H T>
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and possible diffdrent ’Stecker values’ of T and W. Any
assumption as to the Stecker values of T ehd W actimcx Implies
two rod< pa irin gs, and when we have set these rods ur we can look
round <nnd see if the^e are any other Steoker which "re consequenoes
of the rod pairings "nd the Stecker we hpve "lre-ay. Any new
Steoker we find nry allow us to set up more pairs of rods. So we go
on until either no new consequences oan be drawn (this may be rather
frequently the oase), ot there is a contradiction. If there is
confirmation and afterwards we can dr' w no km further oonsequenoes
it may be worth while bringing in extra hypotheses ,
In the actual working it seems best to set the orib out as In
Fig k"L , so th et the occurrences of any letter can be spotted at
onoe. We write the Stecker values of the letters in penoil on the
right? possibly on a separate sheet which slips underneath. In 6rder
fiA *«*(■
to avoid bogus oon ty d i e t lons we oover up thje const^tation^ with
shirt buttons as th^ey are used. Fig 4 . 2 . shews thja working for the
correct hypothesis W/E,T/B. The ’oovered’ letters are shewn ringed.
In order to Shew how the \vorking was done the steps h"vebe"n
numbered, the number being put against the con_ptatRtion used and
also against the Stecker values or rod pairing which resulted.
The work as shewn is not quite complete. It is possible to go
further and get the Steoker value sof "ll letters except D,X . There
are six or more confirmations.
There are a number of other possibilities besidesjworking from
a half-bombe. It depends largely on the number of Stecker expeoted
whioh will be the most profitable. When the number of Stecker is low
(say 6) it is probably best to try helf-bombes as unsteckered and
to look for oliok3 which h've all four letters unsteckered.
It seems unlikely that this method will ever be applied, Partly
because of the difficulty of obtaining the right kind of data.
However th* much the same raetSol d oou2l’%e m appl5e^ith data of the
kind that arises with the air enigma. XkjttRxOn_e may find the
Ringstellung by Herivelismus , and also have a certain number of
oonstetions at known window positions arising from CILLI s
The wheel order may also be known from CIT.TJs more & or less
aocurately. it We now make up rods giving, not the effect of going
through the R.H.W. butjthffough all three wheels, and with the columns
not corresponding toihs all possible positions, but to the
positions where there are known constatatidras , and use them
instead of the ordinary rods: there is no difficulty about T.O.
Identification of wheels
-a
£?« .
When one has found the connections of a wheal one natuffiaily
wants to verify that it is not one of the wheels used in some
other known machine. A convenient way of doing this is to
class of
find th_e ^substitutions which transforms one column of the
rod square into thjs next (see p KfaJ) , Thus the class of the wheel
found on p 26 was 13,8,3,2. iHxkhi>rffxsKxic£xCT^faakslixscs:±x This
rod
’class' is independent of what ooint of the srirexi square we take
to be the top left hand corner, and so is an absolute characteristic
of th_e wheel. It even Remains the s^me if the wheel is used in a
machine with a different diagonal. In the case of an Umkehrwslz
we can farm the dess of the substitution consisting of going
throughthe U.K.W, and then sliding one position baokwards on the
diagonal. A list of characteristics for the known maoftines is given
below
K enigma '
I. 19,7
II. 14,12
III. 10,8,5,3
U. K.W. 15,9,1,1
Servloe machine
I. 13,6,4,3
II. 16,10
III. 7, 7, 6, 6
IV. 11,11,2,?
V. 9, 9, 6, 2
VI. 24,2
VII. 12,5,5,4
VIII. 24,2 two apart
U.K.W. A. 9, 8, 4, 2, 2,1
B. 10,7,8,1
C. 13,8,2,2
?
3
Railway machine
I. 24,2 two anart 18,5,2,1
II. 12,8,4,2
III. 14,8,3,1
U.K.W. 24,2
Commercial
I. 18,8
II. 19,7
III. 12,9,4,1
U.K.W. 22,2,1,1
two apart 9, 8, 6, 3
22,4
CftSpter IV. Single-wheel processes . (Unsteckered Enigma )
We no suppose that we know the connections of the maohine,
and that there are no Steoker, This practically presupposes
that we hr-ve already read some of the traffic, and therefore
that we know something of ttua probable words ±xxx± used, especially
at th^_£ beginnings and ends of th v e messages. Suppose then that
we think that a message storing FKSjSIfQlfxXBBP^KX beoomes when
deciphered DANZIGVON. * . Weshall h°ve to take several independent
hypotheses as to which wheel is in the R.H.B. position, unless
otl^er messages for the day have already been solved. Let us suppose
that the purple whe r l is on the right, We(shall then have to fcske
26 separate hypotheses as to what rod position the xii message
starts in, WtenxwsxsraxtaEytHgxoHtxttexhyPBtkTestacxldKct' ihsx
with the rods, and when trying out the hypothesis that the
pre-start is at 26 on the rods we piOck out therods starting
F F
with F and D and lay them with D under the D of the message
and crib as in Fig 4 ^ . We find on the rods at position 4
enciphered as W instead of I, or else that there was « turnover
between the D ehd the Z , As we do hot think this letter
alternative very likelyjve go on to the hvnothesis that the
pre-stsrt was at 1, and this also gives us a contradiction oh
else a T.O. So we go on until we try rre-st«rt at 4, When we
set up the peir of rods that gives S we find that it also gives
us v,/and when we set up the pair giving I we get also 0, This,
mackacx together with the faot thet there are no contradictions,
makes it praoticelly certain that we have found the right rod
start. We oan then decipher a few more letters of the message,
assuming th ere was no T.O. In this way we get
tt, M
DANZ IGfV ON .ANN tfft. suggesting the decode DANZ IGV ONMANNHE 11,1 , . .
with a T.O. jww m hw e between the aee ond H nnd 1 . ho
. We write the message out in guage
$ which implies th at the Z of DANZIG should have been
J*
iV
MANNHEIM . feKBxsiHKBrfciyxiiBfBXKxihH In order to decode more of the
message we afraid* bn — nfru oan K±±i»x try using the three
couplings after the turnover to read e little more. This is shewn in
Fi l+>> / . It Is not possible to fill in the intermediate letters
and we have to find some oth er method. One is tojtry decoding after
the T.O, with various assumptions about ifes which wheel is in the
middle position, and what rod position the M.W, is in. We shall not
actually need to do the decoding for each suoh nosition, ps a
very large nrooortion of the nossibillties is Immediately
eliminatdd by tb^eflM a tlxltax e known to oocur after the T.O.
In fact we have the seven couplings kUjen.fty^njay.td ,vhj[before the
ftS.
T.O, and tfee itarasx -fcw« o« , le after it » w a- «oa r. i V ly - iso - iw .
We gxnjcr g could treet these couplings with res^eot to^the middle
whe*l in the same way as we treated the original crib with
respeot to the right hjand wheel. However it is not really neoesssry
to get out th_e rods. It is easiest to work with th^p rod square
and for each possible position of the middle wheel look snd see
what ooupling before the T.O, is a oonsequence of oa after the T.O,
For example there are tha bits of red rod
1?
BA
TO
and therefore if tKe message starts in rod position 1 for the
middle wheel the ooupling rav must hve ooourred before the T.O.
(^colour
in order that oe may oc^ur °fte- it. Consequently this position^
for the middle ’ heel is impossible. That the middle wheel rods
can be used in thi3 amounts to nothing more than that they
can be used in decoding in the way described on p. 14,15. In this
way W' find that the only possible positions foi^the middle wheel
is
^r£xx®di siaxxr®±x8xa:nii x saabc xxsxxxtoxxi red 11, and we have
for couplings after the T.O. yg,uv,kt,hh,ws,ora,el,os
from th e first to the second T.O. reads
VKXUZ&i RBZOpVfTKVLDKSNRDBS
EIM.GAN .A.MEETOTER. IT. . ,E.
M
We oan fill this in to rend, forthe whole, message up to thispoint
tairij^iimMxxniD5±BKX DANZ IGV ONMANNHE IMKGANZARMEETOTERB ITTEBEFEHL ,
The other oouplings *xy rf,jz,qi can now be read off the filled
altogether we now have
in letters, and the oouplings of the M.W. rods
qo,er,ab,sx,vra, jm,^t^£l,yd* z l»h!fr. We oan deoode as described in
Chap II ; the two remaining middle wheel couplings will soon be
found.
We might of course use either the middle wheel oouplings or
the righ_$ hand wheel oouplings to find the position of the
L.H.W. and U.K.W. and we oould then do tl^e decoding on a machine
instead of on the rods. Methods for doing this will be described in
the next flhapter. The rest of this chapter will be devoted to
methods of brightening up the *xx first parts of the prooess.
The inverse rods
Instead of pioking out the H?H.W. rods and laying them against
the orib as in Figs 43,44 we might write down the rod
oouplings which are oonsequenoew of e«ch of the const a tat ions, thus
wheh testing pre-start 26
FKSJJBtQtfr
DANZIGVON
omuq ifcijis
wjsonm^Y^
The contradiction which we found before by setting up the pe ir ow
n^pw shows itself in the form of two oontrediotory oounlings
ov;,oq. In the oase of pre-st8rt 4 wp have
FKSJTTQJfY
DANZIGVON
uptlcjfcufcy
kedwfnkra
and our confirmations (clicks) show up as repetitions of the
couplings uk,^f. If we actually did tb^is we should lose time
in comparison with the original process, but -e can eotually get
all the couplings in the different positionsjby a more neihanical
method.
We have the lines of the inverse square (p 10 ) written out
on rods in double length, oalled ’inverse rods’. We
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pick out the xm ix inverse rods n^ned nfter the letters in thj?
crib, end ley then down in pairs, stag ering them b^ckw^rds. This
is best seen in Fig 46 . The verious columns in this set-up
show us the verious rod couplings which ere oonsequenoes of the
orib end verious hypotheses ss to th e pre-start. In the figure
the pre-starts h ve been written along th-je top, but this is not
part of the ormal routine. With this method we can easily see
contradictions whioh are independent of where the T.O. oocurs
e.g, for pre-start 1 we have the couplings wi, wl,jl arising
from the orib in that order. There must be a T.O. between the
wi end th^e wl and also between the wl and Jl, which apart ftj$m
double T.O. is impossible.
Masks
Th ere is another
MBlfe method which gives essentially th e s^me result as the
and
inverse rods and seems to be a l little quicker* to require rather
less permanent apparatus. We need to have th e inverse squares
written out with part of the beginning of the square repeated
<lruL
ag^in at the boginnia g, end in rather small letters. In order to
work a particular orib we take some pap(i^) in g|Q^ge with the
inverse oblong and write the diagonal down th^e side of it, and
EaiESJtxx write the orib along the bottom. Then for each letter of
the orib (either code or deoode) we punch a hole diract in the
column in whioh it occurs, end in tl \jq line neiped after it
(Fig 47). We then move this mask over the inverse oblong. Each
p^Jition of th e mask corresponds to a different st»rt on the rods.
Th e pair of letters shewing through the two holes in « column
give th e ooupling whioh is a oohsequenoe of the constate tion written
in th at oolumnfFig 48).
Another advantage of thid method is that we can test all
colours with one mas£. This advantage can however also be got by
making inverse rods with all the colours on one rod.
7 '
Cherts .
When w- went to try the same dedode for !
s greet many
different messages, end perhaps for many different nlaoes in the
to
same message it may be worth while A make sueoial statistics for
that crib. We o°n make statistics of the positions in which there
will b e 'olioks'. There is quite a problem as to the farm in
which the statistics ought to be presented. I will describe two
forms which have actually been used; named after the principal
cribs for which they were made. First however I must explain
the terminology I shall use. Let us tpke for example the crib
XBRij ESSELXX fitted onto a pert of the message AEIRCMTWBZ J . There
is s cliok as shewn below
19 20 21 22 23 24 24 26, 1 2 3
aeircmt/wwbNzj
X B R U E^S S / E \ / L ) X X
N V Y L C 0 T W BfP U
D G G K W C U \e/\L/ A B
rod nos it ions
message
crib
rod
rod
**'* U.0,
As the constats tionus of the cliok are consecutive I shall say that
the 'olick distance’ is I. W is oalled the 'first oinher letter'
and B th_e second cipher letter, E the first and L thj9 second
'crib letters l As the first letter of the crib comes at rod nosition
19 we Kxti say that the 'rod start' is 19. As th e first orib letter
E is(the eigtoth letter of the crib we say that the crib nosition
of the cliok is 8.
PERCOMMANMTE /charts .
This is the oerfeot form of ohert for use wh^en the po= ition
of the crib in the message is known exactly. The olrjrt has several
major divisions according to the different possible first crib
letters. Each of these major divisions is further divided into
lines labelled with the seoo^nd crib letters, and oolumns labelled
with the first cipher letters. In the xhh ±iuox resulting small
1 V
rectanjLes r- re written the seoon d cipher letter and tie rod start.
Thus th e eighth mpj or division of a PERCOMMAHDANTE tyne chart
made out for XBHUESSELXX would look like thid
A B C . . . W . . .
E&, B 19
e %: 2
E 2 ! 3
all entries apart from the one corresponding to the oliok shown in
nu.wvben
Fig 4 ^ having been omitted. The l s ttor» written above °n t tothe
righ t of the letters in the r^ames of the rows distinguish between
different occurrences of the s r me letter in the orib. By writing the
message downward in guege with the lines of th e ohart it is very easy
to see the possible dicks. We note down th^e rod starts, and, if
we find one of th^em repeated try it out by th e method described at
the beginning of the ohepter.
BRUE5SEL type charts .
These have the advantage over the PERCOMMAEDANTE type ohatts
th^pt one oan investigate all possible pg^itions of the crib in the
message without doin^g them all in _dependently, but it h j>s some
jtcamB nx s niixgxtix counterbalancing disadvantages. In the form in
whioh they were made for the Railway traffio "'ll three colours
were put aarfcHxa together and there were senarate sheets for the
different click distances. I now think that it might be better to
separate the colours and to have three or four cliok distances
on a sheet. In any case the sheets are further divided into
lines according to the different first o toller letters and the
entries in the linesxre oonsist of the seoonjl* cipher letter,
the rod st rt and the orib position of the olick. Thus the oliok
shewn in FlgJtq would be represented on sheet I in line W by the
entry B 19 8 in green. The ohart is usually used one sheet at a time ,'
7 *
the message is written out with plenty of room for entries below it.
Whilst using sheet I sraxjtBOk for e n ch letter of the message wg take
the corresponding line of the sheet and look in it for the letter
which oomes next inthe message. For eaoh suoh entry that we find we
ms kxximxExX ryx srHxtjot enter the rod start on the message under the
letter which corresponds tothe first 3txafxfik6ffi letter of the crib.
We know where this is beosuse the entry on the ohart gives the
orib position. When we get toes the same number twioe in a oolumn
out
we try A the corresponding rod position and position in the message,
A possible improvement of the lay out which might combine the
advantages of the PERCOMM&NDANTE and BRUESSEL type charts would be
to take a fairly wide columnfor each click distance, all the
columns being th e same width, KHixtaxhicx* instead of having
separate sheets, and to make the lines fairly deep. The message oould
then be • ritten out in gujage with the chart. However I am afraid that
this might kack both chart and message un_wieldg. AHatherx-rossibiBrc
±mprsrvem«HtWBia±ibdD«x±oAn alternative possible improvement would
c ipher
be to h s va separate oolumns fcr the different second^ letters ,
This would also mean having rather large charts, because of the
great variation of the number of letters thst would have to gointo
a reo tangle.
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Making of o harts
Although there is so much room for variation in the form
whioh a chart can take the manner in which they are made is
fairly stereotyped. Th ere are two kinds of oliok to be catalogued,
calded' direot* and ’oross*. Direot olioks arethose in whioh both
letters of the crib ooour in th e same rod. Both olioks in Fig 44
are direot olioks. Cross olioks have one of the orib letters on
one rod and the other on the other,
Whjsn cataloguing cross olioke we make 26 niotures like Fig 50 ,
by writing the crib diagonally and filling up a square with rods,
and finally copying the left lower half intothe right upper h alf
symmetrically across the diagonal. The different piotures
corredpond toMifferent rod starts, Eaoh square above the diagonal
gives us an entry for the chart. The lower letter is the first
cipher letter, and the umer is the seoond cipher letter. The row
gives the oliok position, i.e. with a BRUESSEL type chart the
n umber in the ’index* position. The oliok distance (i.e. the she«t,
with BRUESSEL type) is determined by how far t\ja square is from the
oehtral diagomal; in. the figure the squares corresponding to
oliok distfince III are ringed in penoil. With a PERCOMMANDANTE type
ohart we should not use the diagonals butjthe oolumns. Some of the
squares do not correspond to possible entries, as they could only
arise from rods paired with th emselves. These kk£ squares have been
crossed out in Fig ifD ,
For cataloguing direot olioks we heveto find all oases in which |
a pair of letters on a rod can fit with 8 pair of letters of the
crib, e.g.
Each suoh oase will give us 25 different entries in the ch art,
X B R U E S
rod
%
all with the ssme diok distanoe, rod start and orib position*.
In cataloguing these either in a PERCOMMAHDANSE or a BRUESSEL
chart it is sufficient if we put the second cipher letters all in
simijar positions and only once enierthe regia inin g information,
for each set of 25.
X-c harts
Sometimes one will find messages with about 30# of X's in the
decode. These can be got out by a ’majority vote* method, looking
for the R.H.W. starting position whicfc gives the greatest number of
dicks if we assume the message to say XXXXX all through.
If there are edtually 30# of X's there will be about d£ genuine
clicks between X's per T.O. : there will also be an average of
about 0.5 *i±nc apparent clicks arising from letters which are not X,
giving altogeth er 2,7 clicks per T.O. with the correct start. With
the wrong start we have one bogus click per T.O. If we do not
kn oW where the T.O. is these figures have to be modified. In the
righ t olroe we have 3,7 clioks per length of 26, and in th e
wrong place 2,0.
SHdjnaMxkyxTiring. With X-oherts there are less variables
involved than with ordinary charts, as th ere is no question
as to wh ere the orib should be set against the message. The
variables involved therefore are the first and second s±±skx±s
cipher letters, the click distance, and the rod position ofthe
first oonstatation of the cliok. There are two ways of setting the
ohart out, one favoured by Kendriok and one by Turing.
With Turing's form of ohart tfrgxfix stxa infra there are 26 lines
named after the first oip&er letters and 26 columns corresponding to
the possible click distances. The seoond duller letter an d the
rod position are entered in the square. The ohart c*»n be used by
writing the message out in gauge with the ohart, *>nd nutting
each letter in turn over the corresponding letter in the left -hand
V
column which names the lines, and looking for eeoh letter Hi
among the next 26 of the message in the square of the ohart
direotly below it.
th e implied rod start of the message by subtracting the position
in th e message of th e first oipher letter from the rod nos it ion
of the first oipher letter, i.e, the number in the square. We
enter against this rod start th e position in the message of the
first cipher letter. The rod start with the greatest number of
entries against it is presumed to be th e right one. To £eh± read
the message after we have found th e R.H.W. rod start we can
try setting up th e ro s giving the olicks end see if this results
in any further iden tificetions, but this hardly ever gives th
solution, Ax The generally aooepted method is to take xxaac^HXtty
XH t H x the couplings giving the olioks and note down from a
catalogue the places in whioh they could occur, and then take a
’majority vote’.
In mekin g an X-ohart we can make a set-up like Fig W , This
will measure 26 x 26 end x±i±xdBrf«xxx±± only one of them will be
needed. It will simply oonsist of p rod-square rearranged with the
X.*s down th e diagonal. When making th e entries far e particular
rod position of the fifst constatetion of theolick (i.e. th e
entries wh ere a particular number is written in the square )we copy
down a line from the rearranged rod-square, stprting immediately
after the X, aoross the top of the rod square, and also the oolumn
starting at the same X, grab The entry to be made inany oolumn
then
oan be seen by looking et the top. Having made these entries we
In noting the click down we anrtarxxjflrac calculate
value of
in penoil
rub out the line* at the top and replace them with oth ers
7f
In Kendrick’s type of XQchart the intrccH names of the lines
the first
give ±tac bb« of the cipher letters/ The columns give the oosition
seoond oiuher letter and
of the other oipher letter, and the entry in the -qua re is the
position of th e first oipher letterpciraixfcfcK. This form of chart is
particularly useful when we have a hunch about the rod st°rt.
Consecutive tables .
In the seoond part of the process, where we are finding the
position of the middle wheel we can speed up the work by the use of
conseoutive tables. These are cf two kinds, forward and badkward,
and look very like rod squares. The letter in oolumn 18, say and
row R of the forward oonseoutive square is th e letter whioh
occurs In oolumn 19 of the rod with R in oolumn 18. The letter
and row R
in oolumn 18 of the backward conseoutive square is th at whioh
occurs in column 17 on the same rod. Like rod squares and inverse
squares these oonseodtive squaares 'have a diagonalli.e. oan be
?ii5ecl*fn*Frora a single upright by writing *the diagonal* diagonally
downwards toth e left. In our DA1TZIGV0N example we oould h ave
used the baokward oonseoutives as soon as * we hstf found the
oouplings ku,ep,fx,qn,ay,td,vh,lw before the T.O. and sw,oa.le
after it. We should hav laid rulers against the lines o,a of the
backward conseoutive square, and re*d off the oonsequences
before the T.O. of havin g oa after it, in the various possible
positions of th e middle wheel, and would have looked to see
whether th ese consquenoes were consistent with oufc data. We
sh ould then have repeated with ws nnix±hKxriH±±±iix looking only
at the positions consistent with oa. The forward oonseoutives oan
be used wh en the place has been found for reeding off the
oouplings after the T.O. (although this is only a small advantage),
or in a oase where we have started from the end of the message amd
worked backward s
Qo
Chapter V . Coupling catalogues
When we have found the rod position of the R.H.W. and a few
couplings for a message it is possible to find the postions of th
other wheels frofci a suitable catalogue.
Short catalogue
On e method is to try independently allthe possible positions
for the middle wheel. We shall want to know th e middle wheel
couplings whioh are oonsequenoes of these various assumptions.
This oen be done by setting up inverse rods for the middle wheel.
The rods are paired off according to the R.H.W. oouolings, i.e.
M.W. oatput, xxxiaxilgxxxx.This has been done for the the couplings
the red wheel in the middle. The pairs in each column of thacie
set up give possible M.W. couplings. We have nrntx now to find out
whether these couplings are possible. Our procedure is rather diff-
erent according as the U.K.W. does oi? does not rotate. In the case
that the U.K.W. does not rotate iit will be sufficient to have a
(the rows and columns lettered preferably with the diagonal alphabet)
Foss sheet w it h in whioh, in the RW square tAm are entered tne
position s of the left hand wheel at whioh the pnanUng RW is
the ’short oetelogue 1 for this wheel. To use it in connection with
th e DANZIGVON crib we should take each column of Fig ‘>'5' in turn
and look up the pains in it on the short oatelogue<=nd see if all the
squares had a number in commotfu If we found such a case th e number
in the square would give the L.H.W, rod position, and the column of
xtfc Fig would give the M.W. position. Aotually the U.K.W. rotates
for our example so that we should have no suocess.
In th e oase that th e U.K.W. rotates we need essentially the
same short catalogue, but we arrange it slightly differently.
In stead of th e lines of the catalogue corresponding to fixed
output letters they correspond to fixed distaoes on the diagom
illustrate suoh a catalogue. The pairings are written above the
ku,fx*ep which arose in the DANZIGVON orib in Fig b'tf , assuming
one of the pairs in the L.H.W. output
between the outout letters. This may be seen from Figs ^ *'3 which
figures giving the nos At ions
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of th e L.H.W • In which these ppirings occur, the U.K.W
understood to be in the zero position . Either form of short
catalogue mey be made by setting up the L.H.W. rods paired
according to the U.K.W. es in Fig and enelysin : the resulting
pa irs .
To understand the use of the sh ort catalogue when the
U.K.W. rotates we must Matin remember that if the UjK.W. and
L. H.W. are rotated in step the effect is a top slide along the
diagonal of the resulting nairs. If we are given actual nairs
for whioh the U.K.W. was not in the zero position we oan slide
the pairs along the diagonal until we heve pairs whioh would have
ooourred with the U.K.W. ii the zero position. This will show up
number
on the catalogue because there will be a iattxr in common in th e
squares under these pairs ra. For instance in th e case of the
DAUZIGVON crib we found the middle wheel to be inxnux red in
pr,ve,hn,uy
position i*. This gives the middle wheel couplings Kmfijcpiaaprij
as consequences of the R.H.W. couplings qn,uk,fx ( ep . These oan be
read off from Fig although of course we should only set up the
M. W. inverse rods in e case where we did not know the M.W. oosition.
If we slide Mqcra^iiipijtfcpr , hn , e v , uy ten nieces forward
along the diagonalwe get wg,mi,zf,ke, and in each of the squares
wg, mi,zf, ke on the green (L.H.W.) short catalogue we find the
number 4, i.e. these pairs occur 8t U.K.W. 0 L.H.W. 4: consequently
qn,... occur at U.K.W. 10, L.H.W. 14. Th e mechanioel process
would aotually be to take pr on the small sheet of the catalogue
and l^y it against ve on the large sheet. This automatically results
in wg end joi being together and all other pairs of pairs resulting
from sliding pr.ev along the diagonal. We look in the pairs of
squaresto see if there are numbers in oommon. When we find such a
case we have to loolf in a third square resulting from sliding hm.
It is as well therefore to have rulers in gauge with the
catalogue to measure off the distances. Having found the righ t
amount of slide forward on the diagonal, i.e. to th e ri ht in the
catalogue we calculate the positions of the wheels from the fcrmulae
Q-'i 2 X
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U.K.W. position - slide forward on diagonal
L.H.W. position - number in square slide
The Turing sheets
The short catalogue should work very well when the Umkehrwalz
rotates, and there is no information sdBHB connecting the postilon
of the U.K.W. with the positions of the other wheelspcixxfci®. In
the oase of a fixed U.K.W, we oan often make use of an analysis
of R.H.W. couplings.
The lay out of th e catalogues is largely determined by the
special method xfaiBhxig by which they are made, but it seems to be
reasonably convenient in use. The catalogue is divided into sheets
numbered 1 to 13. Each of these sheets oonsists of a 26x26
square with margin at top 8nd left hand side, preferably on l/3 n
gauge.
a. One such sheet is
shown in Fig partly constructed. The letters and numbers in
ink are the only ones concerned when the sheets ere being used, the
others being part of th e construction, end left on to help in
tracing errors. The entries 10,18 ,'21 in the square in column
oounling
15 an d the row with KV in the margin mean that the rnrirr KV
occurs when the M.W. is in position 15 and L.H.W. in any of the
positions 10,18,21. In order to find the positions at which two
couplings oan occur we have only to find the corresponding lines
of the catalogue against one another and oompare the numbers in the
adjacent squares. It is fairly easy to find the right sheet
b eoause the number of the sheet gives the distance along the
diagonal of the two letters of the pair, e.g. K and V are at
distance $ along the diagonal (KFYXCV)and KV oocurs on sheet 5,
pf
Construction of the Turing sheets
The construction of the catalogue depends on making almost
simultaneously all the entries corresponding to gaetr axfg r oases
in which the ourrent flows through the same two wires of the M.W.
In th e partially constructed sheet 5 in Fig (Tt some of the
diagonals have been filled in fully, and each of these corresponds
to a pair of wires of the M.W. As the M.W. rotates the rod
points at the right hand ends of th e wires move steadily
backwards along ’the ddagonel* . We see Irtracfcxiiiixx also that as UfC-
move along the filled in diagonal the rod position steadily
increases, and the letters in the pairings move slide backwards
along ’the' diagonal’ . Meanwhile the left hand ends of the wires
are steadily rotating, so that the middle wheel couplings ate
sliding alodg ’the diagonal’. The entries in the squares are the
positions of the L.H.Vv. where these M.W. couplings cen occur, and t
the slide along the diagonal amounts to a diagonal movement along
the short catalogue. Take for instance the dfff filed in diagonal on
Fig nearest to th e central diagonal. The second entry onthis
diagonal is 2,5,16,26 which is the entry at HL in Fig : next
along the diagonal in Big 0% is the entry 10 which occurs at GM
in Fig 6 7 , and so on , the diagonal in Fig 6"/ being repeated
backwards in Fig n .
This phenomenon may a^^ be explained with reference to the
rod square, instead of the wheels: this is really more practical,
as we have to make the catalogue up from the rod square. A possible
method for making up the catalogue would have been this. In efcch
scuare on th e sheets we write in, in pencil, .the M.W. couplings
which would be needed to prodtioe •fehe/M.W. ouput x ' a quii - e - d at the
A
M..,. position given by the/oolumn In which th e square oocurs.
To do this we should have to - rite down in e n oh line the inverse
rods named after the letters at the beginning of the line. This
has been done in a part of Fig & (ton .H oorner). We should
then have the square filled with one inverse (M.W.) scuare, with
top and bottom reversed, an d another suoh reversed square somewhat
displaced upwards. The entrde in green ink could be obtained by
*1
replacing each pair of pencil letters by the corresponding
entry on Fig V l , i.e. by th e position oS the L.H.< . at which
that pair of letters oocurs as L.H.Y7, output. Now the whole of the
pencil square can be obtained from its to - line simply by filling
in along diagonals. Translated into terms of the green ink entries
this means to say that we only need to be given the oositions at
to
which th* start copying from the short catalogue.
Actually we copy out the diagonals of th e short catalogue
onto staircase shaped strips (known as 'Christmas decorations' or
'hand frills' ) in reversed order, with the position in th e
short catalogue written above each square. These h^nd frills
8 re numbered by th^e (constant) d 1*
in position for copying in fi Fig I and F are at distance £ zS
on qwertzu and so are D end K, Insteed of actually filling ;in the
whole square with pairs of penoil letters we take the entries which
migh t have been maxd.e in the too line, an d write them in the top
margin, •~nd also macks put th^e entries which might have gone in the
left hand column into the left henri margin. In order to find what
h^arurti frill to use for a particular diagonal the distances apart
alon-g qwertzu of the letters along the too are calculated. This
should be done quite independently, to give a check on incorrectly
copied letters (see 'MysjJio numbers').
The reason for having the imaginary pod squares implied in the
construction inverted is in order that the writing of diagonals
may be from left to right and downwards, which is considered easier
than from right to left and downwards.
of tlv-e pairs of letters on them;
K
L
M
S olving a short crib
The isc&xi oh ief application of the Turing sheets is to the
when th e U.K.W.does no rotate
solution of cribsfrom a length of 8 to 6 letters. We set up the
inverse rods as usual, but find th8t xx xxivxxsciixihx by no means
incorrect
all th e KHirsai positions ere eliminated by coupling contradictions.
We therefore look to see whether there Is any oosflition in which
th e couplings can occur. Take for example th e crib ANX, ’-'ith
and wheel order I III II (red, green, ourple), U.K.W. nos, 0
cipher »*£. We se-fc U P th e inverse rods »s in Fig gj , an d for
each oolumn of the resulting set up oompare the lines of the
catalogue named after th e pairs in the column. For each pair we
shall wnt to find quiokly th/-e right sheet on whioh to look, and
this meand subtracting the pairs on the diagonal (i.e, finding
their distance apart on qwertzu). To do this we can either have a i
table of differences or else use 'mystic number rods*
' lWstio numbers '
Fig shows a table of 'mystic numbers' for the red wheel.
The meaning of the tabl is this. Take the 8th line for e---mole.
It could be made by tpking rrat inverse rod Q end inverse rod 0,
qwertzu
0 being eight places on along tin xAxj c jm xA from Q. lay the two
rods together end find the differences of th e resulting ueltrs; e.g. j
fifth
the tkjbni entry in line 8 is 6, *>nd the fifth letter of th e red
inverse xjpoxxKxxx rod Q is Y, the fiftth letter of inverse rod $ 0
6
is F, and Y and F ere fixaxepRrt on qwertzu (FGHJKPY) , If then we
had a set up of inverse rods including the pair QO we could use
tell
the series of numbers of line 8of the mystic n umberw to g±xa us
on whioh she ts the various pairs should be looked up. Hov:ever we
oan also use line 8 of this table on many other occasions. Suppose
for example that the pair ES of inverse rods is up. The seties of
sheets on whioh we have to look is again iven by line 8, but we
have to start in th e third column under E instead 6f at the beeinnin
under Q. Quite a oonvenien t ar angemen t is to h*=ve the lines of
thS table written out on rods in gauge with th e inverse rods and
of double length. (This was once done for the service machine wheel
III, Three lines of th e table were put onto acxxh three sides of
Mr Knox's blank wooden inverse rods, n nd the fourth side occupied
with the letters of the diagoanl, in that o'se XEB A BCD... It was
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not a eucoess as the rods v/ere inoorreotly copied). For the crib
7*
BRC
A NX th. ese mystic number rods pro shovm in position over the
inverse rods in Fig ^“7 • Every fifth letter from the ton mar of
the mystic number table is also shown.
Another use for the mystic number table is in the making of the
Turing sheets. The line of penoil numbers along the top of any
sheet is the lin e of mystio numbers with th e sheet number as its
lime number, and starting at column L,
The mystic numbers orn of oourse be made by actual subtraction
from the inverse rods. However it is actually easier to vntisxxX.1 is
aab txiTBliKK do the oeloulation in terms of tie letters of an
upright. It turns out th^-at one can manage with orue upright, which
one subtracts from itself, staggered various amounts. One oan
tiD-BisfHXB transform th^e letters into numbers sms to simplify the
subtraction, Ishall not give th^e details of this.
EINS cat- logues
In this ohapter and the l^st wa have not exh°usted *11 the
possible methods of dealing with the Unsteolcered enigma, and enigma
with known Steclter, When the Umkehrwalz does not rotate we c«n
catalogue the result of encoding a short word such as EINS at
every pos ible position. The details of this are explained in Chapter
*
Jeffreys ah eets
In oases where the wheel order Is unknown it is useful to
have *±ixfciiE the positions sdraxs: and wheel orders where a ooupling
occurs all catalogued together. In order to make comparison of
couplings feasible one puts the catalogue into the form of punched
sheets, which can be laid one on top of another. These are known
as Jeffreys sheets.
T he actual form of the Jeffreys shots catalogue is this. There
are 325 sheets labelled AB,AC, . . .AZ,BC, . , . ,BZ, ,YZ. Each
sh eetx measures 26"x20 4 /5" plus margins of about three inches.
They are divided into xxxfct columns an inch wide, and lines 4/ 5 "
1 " 1 "
deep. The whole is further subdivided into squares /5 x /5 .
The /5 x 1" redtengles correspond to the different possible rod
positions of the L.H. and M.W. The subdivisions of the rectangles
correspond to the twenty pos ibl wheel orders for L.H.W. and M.W.
with the five first wheels of the service machine.
t
Jeffreys-Turing sheets
There is a possibility of speeding up th e work with short cribs
wh re the U.K.W. rotates by making the Turing *heets in nunohed form.
Suppose we expand every square of the Turing sheets into p rectangle
7/5"x4/5" divided into 28 smell squares, numbered 1 to 26 with two
unused, and for each entry on the Turing sheet punoh a hole in the
corresponding small square. Then th e effect of laying two of th e
sheets on top of one another, in such a way say that the lines
VM and CR coincided would be to give ±s us the positions in whioh
the two couplings VM and CR occur when th e U.K.W. is in the zero
position: we also get the positions in which the couplings sax
slid along qwertzu oocur: but these after making a oorreetion for
th e amount of slide are just the positions at whioh VMand CR oocur
inolmdlng all possible rotations of the U.K.W. One would presumably
normally place three sheets on ton of one another, and there would
have to be four different leyings (because one could not have the
she ts in cylindrical form). For this reason it ”ould be better to havi
the sheets in double depth, but this would probably be out of the
question.
A. Afc-. U*s*r | Ck. u^^jL Q—
p~r.u>
Chapter VI, The steokered enigma, Bombe and Spider.
liVhen one has a steckered enigma to deal with one^ problems
neturally divide themselvBs into what is to he done to find the
Steoker, and whwt is to be done afterwards. Unless the indicating
system is very ’-'ell design ed there will he feo problem »t all when
the Steeper hjave been founjl , pnd even with r good indicating
system we shall be a^le to apply xzmj?±aroclns3c± th e methods of the
last two chapters to th_e individual messages. The obvious example
of a good indicating system is the German Naval enigma oipher,
which is deg.lt with in Chapter VJL, . This chapter is devoted to
methods of finding the Steoker. Neturally enough we never find the
Steoker without at the same time findin much other information.
Cribs.
The most obvious kind of data for finding the keys is a ’crib* ,
i.e. a message of which a part of the decode is known. We '•hall
mostly assume that our data is a crib, although eotuelly it may
be a number of cohstatations arising form snoth er soured, e.g.
an number of CILLIs or a Naval Banbarismus,
FORTYV/EEPYYVJEEPY method! .
It is sometimes possible to find the keys by pencil and paper
methods when the number of Stacker is not very great, e.g. £ to
One would have to hope that several of the constate tions of the
crib were ’unsteckered' . The best ohsnee would be if the same
pait of letters occurred twice in the crib (a 'half-bombe * ) . In
this case, assuming 6 or 7 Steoker there would be a 25# chance of
both oonstatations being unsteokerdd. The positions at whioh
these consta tions oocurred could be f'und by means of the Turing
sheets ( if th ere were three wheels) or the Jeffreys sheets. The
positions at which this oocurred eould be separately tested.
Anoth er possibility is to set up th e inverse rods for the c~ib
and to look for clicks. There is quite a good chance of any
apparent click being a real click arising fram beceuse all four
letters involv-d are unsteokered. The position on the right hand
/ i y
*7 '
" r heel is <jiven by the c61umn of the inverse rod set-up, end we o n
findr.lll possible positi ns where the click coupling occurs from
the Turing she ts or the Jeffreys she-ts. In solne cases there will
be other constatations which are made ua from letters supposed to
be unsteokered because they occur in th e click, p nd these will
furjrher reduce the number of nieces to be tested.
These methods h eve both of them given suo essful results, but
they are not practicable for oases wh ere there are many Stecker, or
even where there are few Stecker °nd rrp ny whe lorders,
A meohanicel method. The Bombe.
Now let us turn to the oase where there is * large number of Steck(
so many that any a- tempt to make use of the ±nctxthxx unsteokered
letters is not likely to succeed. To fix our ideas let us take a
particular orib,
1 2 3 4 5 6 7 8 9 10 11 12 13 14 IP 16 1* 1 19 20 21 22 23
DAEDAQOZS IQMMKBILQMPWHA
K E IN-EZUSAETZEZUMV ORBER I
24 25
I ¥
Q, T
Presumably th e method of solution will deoend on taking , i s|
hypotheses about parts of the keys and drawing whet conclusions ’
one can, honing to get either a confirmation or a contradiction.
The parts of the key_s involved are the xsidxxlxxtxxfxitas wheel
order, the rod start of the orib, whether there are any turnovers
in the crib and if so where, and tl\_p Stecker. As regards the
wh^§ 1 order one is almost bound to consider all of these
separately. If thfc orib were of very great length one might make
h^o assumption about what wh eels were in the L.H.V/, position
an d M.W. position, and apnly th e method we have oalled a
•Stecker knock-out* (an attemnt of thi3 kind w°s made with the
•Feindseligkeiten* crib in Nov. f 39), Donfectn or one might sometimes
make assumptions about the L.H.77. and M. but none, until a ’ °te
stage about the R.H.W, In this cose we have to work entirely
with const 0 tations where the R.H.W. has the same oosition. This
method was used for th e crib from the ingxnf Schluesselzettel of
the Vorpostenboot , with success; howev-r shall assume that all
s
wheel orders are being treated separately. As regards the turnover
one will normally take several different hypotheses, e.g.
1 ) ^turnover between positions 1 -^nd 5
2 )
3)
4)
5)
With the firs of these hypotheses one would have to leave/the
K
constn tat ions in positions 2 to 4 and similarly in all
the other hypotheses four oonstetations would hve to be omitted.
°ne oould of course menage without leaving out any oonst^t^ions
at all if one took 25 different hypotheses, and there will always
be a problem as to what oonstatations oan best be dispensed with.
In what follows I sh^ll assume we are working the T.O. hypothesis
numbered 5) above. We have not yet made sufficiently many hypotheses
to be able to draw any immediate conclusions, and must therefore
either assume something about th_9 Stecker or about the rod start.
If v/e were to assume something about the Stecker our best chanoe
would be to asnumethe Steoker values of A ~nd E, or of E and I, as
•e should then have ika two oonstatations oorreoted for Steoker,
with only two Steoker assumptions. With Turing sheets one could
find all possible places where these constations occurred, of
which we should, on the average, find about 28.1. As th^pre would
be . 52Q hypotheses of this kind to be worked we should gain very
little in comparison with separate examination of all rod starts.
If there had not been any half-bombes in th^p crib we should have
fared even worse. We therefore work all possible hypotheses as to the
rod st rt, and to simplify this we try to find characteristics of the
crib which are independent of th^e Steoker. Such characteristics oan
be seen most easily if the orib is nut in to the form of a oioture
5 end 10
10 and 15
15 and 20
20 and 25
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Fig . From this picture we see that one characteristic which
ie independent of th e pecker is that there must be abetter
which enciphered at either position 2 or position 5 of the orib
gives the same result. This may also be expressed by saying
that th^_ere must be abetter x faig j a suoh th»t, if it is enciph-
ered at position 2, and the result reeno inhered at position 5
the final result will be th^ original letter. Another such
the same
condition is that
letter xk± encinh ered
successively at the positions 3,10 must lead back to the
Three
original letter. Sfcni other conditions of this kind are that
the successive enoipherments at positions 2,23,3 or at
2,9,8,6,24,3 or at 13,12,8,9,5 starting from the same letter
as before must lead beck to it. There are otter suoh series,
e.g, 13,12,6,24,3 buf these do not give conditions independent
kHii The letter to whioh all these multiple encipherments are
applied is, of course, the Stacker v°lue of E. We shall call
E the ’central letter’. Any letter oan of course be chosen
as ’central letter!), but the ohoice affects the series of
positions or ’chains’ fofc the multiple encipherments. There
are other conditions, as well as these that involve the
multiple encipherments. For instence’the Stecker v-lues’ of
the letters in Fig must all be different. SkKxsxiitsirkExxx
values for E,I,M,Z,Q,S,A are the letters that arise at the
various stages in the multiple encipherments and the values
for WjT,V,N,D,K oan be found sin larly. There is al30 the
condition th«t the Stecker st be self-r^oiproosl, and the
other partsfof Fi , P^B-U-O and R-H ’"ill Iso restrict the
possibilities somewh^at. Of these conditions the multiple
encipherment one is obviously th^e easiest to apply, r nd -ith
a crib as lonjs as the one '-hove it - wiH. b e- Puito ouffi -eleat
tO 1—
this condition will be quite sufficient to reduce the aunfean:
possible positions to a number which can be tested by h^nd
methods. It is aotually possible to make use of some of the
oth er conditions mechanioslly also; this will be explained later.
In order to apply the multiple encinhe -ment condition one ,
natu rally wants to be able to perform the multiple enoimhermats
in one operation. To do this we make 8 nevr kfind of machine whioh
we call a 'Letohworth enigma 1 . There are two rows of contacts
in a Letohworth enigma each labelled A to Z end caller the
input itfa and output rows: there are also moveable ’"heels. For
each position of an ordinary enigma the-r-a is a corresponding
position of th e Letohworth enigma, and if the result of
enciphering F at this position is R, then F on the input row
of the Letohworth enigma is connected to R on the output row,
and of course R on the input row to F on the output tow, Suoh
a ’Letohworth enigma' oan be msde iiyxiuixixg like an ordinary
enigma, but ’-ith all the wiring ±iax&Hp±±ExiK^ of the movable
wheels in duplicate, one setof wires being used far the
journey towards the Umkehrwalz, and the others for the return
journey. The Umkehrwalz has t"'o sets of oontaots, one in
unncfcxsjra contact with the inward -journey wiring of the L.H. .
andl one in contact with the outward- journey wiring. The
Umkehrwalz wiring is from the one set of contacts eoross to the
other. In the actual design used there were some other differences;
th e wheels did not actually come into contact with one another,
but each came into contact with a ’commutator’ bearing 104 fixed
oontaots, J 'hese contacts would be eonneoted by fixed wiring to
oontaots of other commutators . These contacts of the commutators
can be regarded as physical counterparts of the 'rod points'
and*output points’ for th e wheels.
one ,, .
If one lias two of these 'Letohwdrth enigmas’ one can
oaonneot the output points of the one to the input points of it
the other end th^en the connections through the t ’"0 enigmas
between thja two sets of contacts left over will give the
effect of successive enoioherments at the positions occupied
by the two enigmas. Naturally this can be extended to
the oese of longer series of enigmas ,BjnxxK*±xot the output
of each being connected to th^3 input of the next.
Now let us return to our crib and see how we could use these
letchworth enigmas. For each of our 'chains' we oould set up a
series of enigmas. We should in feot use 18 enigmas which we
will name as fallows
A1,A2 with the respective positions 2,5
B1,B2 3,10
Cl ,C2 ,C3 2,23,3
D1,D2,D3,D4,D5,D6 2,9,8,6,24,3
El ,E2 ,E3 ,E4 , E5 , 13,12,8,9,5
By 'position 8' we here mean 'the position at which the oonst°t-
ation numbered 8 in the crib, is, under the hypothesis we are
testing, supposed to be ehcirftered' , The enigmas are connected
up in this way: output of A1 to input of A2: output of B1 to
input of B2: output of Cl to input of C2, output of C2 to incut
of C3: eto. This gives us five'oheins of enigmas' "hioh we may
call A,B,C,D,E, nd there must be some letter, which encinh ered
with each chain gives itself. We oould easily arrange to have
all five eh^ains controlled by one keyboard, and to h ave
five lampboards shewing the resultsof the five multiple
encipherments of the letter on the dnreseed key, ifxfcfcaxxsnork
wnnn nrnnrt-iy ^ After one hy othesis as to the rod start had
been tested one would o on to the next, and this would usually
involve simply moving tlB xlgkxxteR .H.W. of eaoh enigma forward
one place. When 26 positions of the R.H. . have be'n tested thja
M.. . must be mane to move forward too. This movement of the
wheels in step can be very easily done mechrlic 0 lly, the righ t
hand wheels all being driven continuously from one shaft, c nd the
motion of tli_e other xvheels being controlled by a oerry mechanism.
register ing
It now only remains to find 9 mechanical method of £e±«ES±ir±ng
whether th _p multiple encipherment condition is fulfilled.
This c^n he done most dimply if we are willing to test each
Stecker value of the central letter th rough out all rod starts
before trying the next Stecker val’ e. la/
o U p ose we are investig-t iing the case sf: where the Stecker value
of the central letterta E is K. We let pi ourrent ent*- "l - * of
the chains of enlgmss at their K input noints, r -n d at the K
output points of th e chains we put relays. The 'on' points of
the five relay s are put in series with r battery (say), and
xxffsxMirfexxtiir-fiD another relay. A current flows throughthis last -
relay if and only if a ourrent flows through all th e othe five
relays, i.e. if the five multiple enoiphj^rments applied to K all
give K. When this happens the effect is, essentially, to stop the
machine, n d such an occurrence is known at Letch’"orth as a
•straights. An alternative possibility is to have a quickly
rotating *scen ner' which, during a -revolution^ ’"ould fi^st
connect the P irrautk of th^ chains to the current suowly, and the
output points A to the relays, end th en would oonneot the
Input end output points B to the sup-ly »nd relays. In e
revolution of tTu> scanner the output and Incut noints A to Z
would all have theih turn, ant the right hand wheels would
then move on. This last possible solution was called ’serial
soanning' and led to all th e possible forma of registration
being known as different kinds of dscsnning' . The simple possibility
th at e fir 7 t mentioned was celled 'single line scanning’,
Naturally there we - muoh research into possible alternatives to
these t 0 kinds of scanning, which would apt enable all 26
possible Steoker values of the central letter to be tested
ni simulteneosly without any parts of the noohine moving.
Any devioe to do this was described as 'simultaneous soanning'.
The solution whioh was eventually found for this ■problem
was more along ma theme tic a 1 th«n along eleotrioal engineering
lines, an_4 would really not have been a solution of the
to
problem as it was put to the e lea trie inns , tm. whom we gave,
as we thought, just tbjs essentials of the problem, It turned out
in the end that we h^ed given them rather less th^an the essentia^
and thjay therefore cannot be blamed far not having found the
best solution. They did find ~ solution of thja nroblem as it
was put to them, which would probably hove worked if they had
had a few more months experimenting. As it was the mathematical
solution was found before they had finished,
Pye simultaneous scanning
The problem as ivjhg, to the eleotrioian^s was this, The e are
52 contacts labelled A...Z, A , ,...,Z'. At ° n y moment eaoh one
of A, ... ,Z is connected to one and only one of A 1 , , . . ,Z f : the
connections are changing all thj? time very quickly. For eanh
letter of th e alphabet there is a relay, and we want to arrange
that the relay for th^e letter R will only close if oontsot R is
connected to contact R’ ,
me* xnsrraTmsxryrercre
The latest solution proposed for this problem depended on
having ourrent at 26 equidistant phases corresponding tothe 26
different letters, Th^ere is also a th^ratron valve*for eaoh
letter. The filaments of the thyratrons are given potentials
corresponding to their letters, and the grids are connected to
th e corresponding point" A*,,.,,Z*, Th e points A,...,B are al90
"*A th^yratron v lve has the property that n^o our-eht flows in
the anode circuit un w til the grid potential xxwxhexxscxnxrtici*
becomes more negative then c certain critical amount, after
which the ourrent continues to flow, regardless of the grid
potential
potential, until the anode p Kxxxricfc is switohed off.
16
t>
given potentials with the phase of the letter concerned. The
result istl^st the difference of potential of the filament and
of thyratron A
the grid osoillates with an amplitude of at le-st
26 chase ■* )
E being the amplitude of the original supply jraingx, unless
A ehd A* are connected through the ch n in, in which onse the
potent ielsre main the same or differ only by whatever g-rid bias
has been nut into the grid circuit. The thyretribns are so
2VU /
adjusted that an oscillation of amplitude will bring
th e potential of th e grid to the critioel value and the
valve will' fire*. The valve is coupled with a relay which
oiily trips if the thyratron fails to fire, This re ley is
actually a 'differential relay', with two sets of windings, one
carrying a constant ourrent and the other carrying the current
from the anode oirouit of the thyratron. Fig (o & shews a possible
form of circuit. It i KXHtx±Ht®nis nrobably not the exact form
of circuit used in the Pye experiments, but is given to
illustrate the theoretical possibility.
The Spider
We can look r t the Bombe in a slightly different way as
a machine for making deductions about Steoker when tl^e rod
start is assumed. Suppose fax we were to put lamp-boards in
between th_p enigmas of th^e ohains, and label the lsmp-boerds
with the appropriate letters off figure . For example in
chain C the lampboard between Cl and..C8 would be labelled A.
■^f we were using the-^ efai ne wjrfch- a key-board tfffc could be
labelled with the 'oentral letter*. Now when we deppeds a
letter of th^ key-board we oan read off from the lamp- boards
some of
th_p anaKxaaptx: Steoker oonsequences of the hypoth __psis that th e
depressed letter is steckered to the oentral letter; tier* for
one such conse^uenod could be read off each lampboard, namely
that thus letter lighting is steckered to the n j * me of tie
lemp-bosrrd.
‘°1
When we look et the Bombe in this way we see that it would
be natural to modify it so as to make this idea fit even better.
We have not so far allowed for lengthy ohains of deductions; the
possible deductions stop as soon as one aomes Vck to th e
cehtrel letter. There is however no reason why, when from one
Steoker value of
h ypothesis about thej^central letter we have deduced that the
oentral letter must have another Stecker v r lue , we should not
go on and drew further conclusions from this seoond Stecker value.
At first sight this seems quite useless, but, as all the deductions
are reversible, it is actually very useful, for all th e oodolusions
that can be drawn will then be false, snd those that remain will
stand out clearly as Dossible correct hypotheses. In order that
8ll these deduetions may be made eehanically we shall have to
connect the 26 contaots at tt^e end of each chain to the
common beginning of all the chains. With this arrangement we o n
of n n enigma
think of each output or incut point »s renr seating
a possible Stecker, and twm if two of these points are connected
together through the enigmas th en the corresponding S- ecker
imply one another. At this point we might see h ovr it all works
out in the case of the crib given above. This crib was actually
xithx^i^BKKXshBDci; «lph abets which , when corrected for their
sSteoker, are those arf -e f - Fig — . , the numbers over th e crib
conste tat ions giving the oolumn o of . The alphabets
below
most \lsed are 2,3,5,10,23, and these are reproduced h ere for
reference
2
3
5
100
23
XN
XH
MD
TB
LV
AP
BU
JZ
IH
WC
EN
CV
RU
DI
cv
PK
SA
XE
OM
TP
QI
YE
CV
XU
UO
AW
GR
JY
ET
MS
ov
PCi
DE
JP
BD
JY
NW
SL
GE
IW
DM
LH
ON
AY
JZ
RZ
BX
Qw-
NB
GR
SL
FU
AZ
HS
YE
GT
01
PK
ZQ
HL
EC
KT
GM
RK
J0$
In Fig £/ at the ton ere the chains, with the no it. ions
ere
end the letters of the chain. In eeoh column ±x written some
of the letters ian which xxxixha: c°n be inferred to be Stacker
values of the letters at the heads of their columns from the
hypothesis that X is a Stecker v°lue of the central letter E.
By no me^ns all possible inferences of this kind are mede in
the figure, but among those that are mede are all possible
Stecker values for E except the right one,L, If we had ta'en a
rod start thet w- s ’"rong we should almost certainly h r ve found
that ell of the Steoker values of E could be deduoed from- any one
of them, and this will hold for eny oribs with two or more chains.
Remember* ing no v- that with oufc ocxtkac Bombe one Stecker is
ded, cible from another if the corresponding points on the lamp
boards are connected through th e enigmas, a correct rod start
can only be one for which not «ll th e input points of the she ins
ere connected together* the positions at which this happens <re
almost exactly those at whioh tks a Bombe with simultaneous
scan ing would hive stopped.
This is roughly thf idea of the ’spider*. It has been described
in this section as a way of getting simultaneous scanning on the
Bombe, end has been made to look ss much like the Bombe as
possible. In the next section another description of the spider
is given.
The spider . A second d scriotion . Actu ° l form .
In our original description of the Bombe we thoughjt of it
as a meth^od of looking, for characteristics of a orib whioh are
independen t of Stecker, but in thja last section we though_jfc
of it more <=s a machine for making Stecker deduction's. This l^st
way of lookitkjg n t it ha obviously great pos- ibi 1 ities , -nfl so
we ill start fresh with this ide 1 .
In th e last s : ctio$ various points of thje cirouit were
regarded as gH Xxgspg mitxgxjao having certain Steoker corresponding
to them, Hxxb We ere now going to carry this idee further and
*
have a metal point for e$ch possible St cker. These we can
imagine arranged in e rectangle. Each point has p such
as Pv: here the capital letters refer to ’outside’ points end
the small letters to 'inside letters’; n n outside letter is the
name of a key or bulb, ^nd so c r n be a letter of a crib, ■ r hile
an Inside letter is the name of a contact of the Eintrittswalz,
so thu't all ±nfBzam±±anx jEi oc t constetations obtained f/G&n 3 m
3QKs±3tKjcBXE±x the enigma without Stecker give info motion about
inside letters rather than outside. Our statements will usually
be put in retK llogical form; statements like *Jis an outside
letter' -ill usually mean ’Jis* eocurring in so end so * - the neme
of a key rather than of a contact of It he Eintrittswalz’. The
rectangle is called th e 'di^gonel board’ end the rows are named
after the outside letters, the columns after hhe inside letters.
a
Now let us take any constetation of ou±x crib e.g. I at 24. For
the position we ^re supposed to be testing we will h*-ve an
enigma set up at the right nos it ion for encoding this
oonstatation, but of course without any Stecker. Let us
supoose it a un for the correet nosition, then one of the
pairs in th e alphabet in position ?4 is OC: Consequently if
a {± ^o then Ic (i.e. if outside letter Q, is associated with inside
o th^en outside I is as ociated '.vith inside c). Now if we connect
the input of the (Letehworth) enigma, to the corresoonding ooints
of the diagonal board on line Q and th^,e ouput to line I then
since th e "o input point is connected to the c outout ooint we
shall have Qo on the diagonal board connected to Io through the
the
Letohworth enigma,
S^eBkBxxxre:cDxxai±sie]ixiKx:BBJOTBC'tinic8X3i
iliHOTnxixbnxxi* ifxHxxthHxdlz We can of course nut in a i>etoh’"orth
enigma for every oonstatation of the orib, andjthen we shell have *
all the possible deductions that oar. be made abo^t th e association
of inside and outside letters paralleled in the connections
between th e points of th e diagonal board. We oen also bring in
the reciprocal xs property of the Stecker by connecting together
diagonally oppositepoints of the diagonal board, e.g. oonneoting
Pv to Vp. one oan also bring in other conditions about the
- . - 1
II o
Stacker, e,g. ifone knows that the letters which were xixiskKxxd-r
XHXXXXxQEgc unsteok red on one day are invariably steokered on the
next th en, having XH±xsd found the keys for one days trsffio one
could when lookin : for the taeys for the next ddy, connect together ,
ell po ints of th e diagonal board which correspond to non-
steokers wh ich had occu rred on the previous d^y. This ■ ould of
oourse not entirely eliminate the inadmissible solutions, but
would enormously reduee their number, the only solutions which
would not be eliminated being those whioh were inadmissible on
every *rawy oount.
One difference fxxmTihsrxXBxks^xxxixika: between this arrangement
and the Bombe, or the spider as we described it in th e last section j
is that ere need only one enigma for each ®->nstatstion.
Our maoh ine is still not complete, ag we h ve not put in any
mechanism for distinguishing oorrect from incorrect positions. In
th e case of a crib giving a picture like Fig 6 r< j where most of
the letters are connected together into one f web* it is sufficient
at some point on
to let current into the diagonal board nx some line x±kh named
after a letter on the main web,e,g. et the Ea point inthe case cf
the crib we have been considering, -^n this case the only possible
positions will be ones in whioh the current fails to reaoh all
th_e other points of the E line of the diagonal board. We can
detect whether this, happens by connecting the points ofthe E line
through differential relays to the oth er pole of our ourrent
parrellel with one another and in series with the stop raecl
supply, end putting the ' on’ points of the relays in sos£±bs. Normally
current will flow through all the differential relays, andthey
will not move. When one reaches a position whioh might be correot
the ourrent fails to reach ± one oj? th^ese relays, and the ourrent
permanently flowing in the other wiring ooil of the relay o c uses
it to dose, and bring the stopp ing mechanism into Play. ihixx
xiiixmxsiiyxfaacppsn with Mostly what will happen is that there will
be just one relay which closes, and this will be one connected to
a point of ths diagonal board whioh corresponds to a ^teoker
whioh is possibly correct; more accurately, if this Stecker is
not correot the position* is not oorrect. Another possibility is
Ill
that all relays close except the one conneoted to thja point
at which the current enters the diagonal boerd, and this noint
xxxttexiixtiraxirarEKaEaclE ±x th en corresponds to the only nos'ible
Steoke’-, In cases where therac data is tath^er scanty, and the
stops therefore very frequent, oth^er things may hj»n-en. e.g
we might find four relays closing simultaneously, all of them
conneoted together through the enigmas andthe cross oonneotions
of the diagonal board, and therefore none of them corresponding t
to possible Stecker,
In order for it to be possible o make th e necessary connections
between the enigmas, the diagonal board and th e refcays there has
to be a good de'l of additional gear. The input and output
rows of the enigmas are brought to rows of 26 contacts c c lled
’female jacks’. The rows of th e diagonal board are °lso brought
The 26 rel- ys and the current supply are also brought to a jack,
to female jacks, A ny two female jacks c»n be oonneoted with
’plaited Jacks’ consisting of 26 wires plaited together and
ending in male Jacks which can be plugged into the female jacks.
In order to make it possible to conneot.±h4 three or more rows
of contaots together one is also provided with ax ’commons 1
consisting of four xhx female Jacks with corresponding points
conneoted togeth er. There is also a device far conneoting
together th e xn?®± output Jack of one enigma xit and the incut
of the next, both being connected to another female jaok, whioh
oan be used for connecting them dcpAkeixt to anywhere else one
wishes ,
On the first spider made there were 30 enigmas, end three
diagonal boards and ’ inputs’ i.e, sets of relays and stopping
devices. There were also 18 sets of oommons.
Figs LI t L If shew the connections of enigmas end diagdphl
board in a particular case. The case of a six-letter slnhebet
has been taken to reduce the size of the figure.
The actual origin of the snider was not xx an ettemot to
find simultaneous scanning for th e Bombe, but ±x to make use
of th reciproo c l character of the Steoker, This oo urred at a
time when it wa clepr that very muoh shorter cribs would have
to be worked than could be managed on the Bombe, Welohmen then
bjr us ng a diagonal board one could get the coraolete set of
consequences of a hypothesis • The ideal machine that Welchman#
was kmi±xxx±ExsK± aiming at was to reject any position in which
a certain f ixed-for-the-time Stecker hyoothesis led to any
direct contradiction: by a direct contradiction I do not mean to
include my contradictions which ean only be obtained by
considering all Steoker values if some letter indeoendently
and shewing e-ch one xmKBxxlfrix iiL-consistent with th p original
hypothesis. Actually the spider does more than this in one way
end less in another. It s not restricted to dealing with one
Btecker hypothesis at a time, and it does not find ail ai rect
contradictions.
Naturally enough Wer iahxmxrix se Ixtsxwaxk WelchmanA and Keen
set to work to fin d some way of adapting the spider so as to
detect all direct contradiction s. The result of this researoh
is described in the next section. Before we can leave the spider
however we should x±xx sea what sort of contradictions it will
detect, and about how many stoos one will get with given d°ta.
First of all let us simnlify th e oroblam md consider x
only itaBxxxxtBxx* norma l' stops, i.e, oositions s-t which b y
' rs the diagonal
current will th en be
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entering at e correct Stecker if the nodition is correot. Let
us further simplify the problem hy supposing thnat th ere is
only one ’web*, i.e. that the ’picture’ formed from the part of
th e crib that is beihg used forms one connected niece, e,g. with
the crib on p we should have one web if we omit the
P B u R
constetations B, U,0,H. mu f fle ion A. u onditi - on £ar "
i s- th th e~* multi - *^ -»_ €>nci ' !»heiment l - o o ndillu i is -slioul^-
liuh l . ~ u| i _ n i y .1 Mf» I fl '"I Ilia 1 1 1 1 ml n 1 1 n I t ll fl i" [i h i i h' Is n I > ■ 1 1 1 , , in
l s ~ J -^ f “ 1 T^.n T 4 'r- 1 1 Mr 'i tih " -v — n f fine -Hv
4-o
i» 1 1. i| ii l ».i i " ti u i i uhojaiaub uuiUT T lluiia h e ld will li u fTJUul UB ^
Some of th e oonstatations of the web oould still be omitted
without any of th e letters becoming disconnected form the
rest. Let us ohoose some set of such oonststetions , whtKfccBxmraik
any more constat* t ions without th e web breaking up. When the
conststeti ns are omitted there will of course be no ’chains*
pr ’closures*. This set of oonst*tetions may be called th e
*oha in-closing const*tation s’, and th e others will be oplled
th e *web-forming const* tat ions’ . At any position we ’ey imagine
that th_e web-forming constate tions ere brought into play first,
and only if thje position is possible far these are th-.e
chain-closing o ousts ta tions used. Now the Stecker value ofjt-he
input letter an^d th^e web-forming con^ta tat ions will completely
determine the Steoker values of the letters occurring in the web.
When the ohain closing oonststetions arebrought in fcka: it will
elreedy be completely determined what are the corresponding
’unsteckered* conste tation s , so that if (there are c ohein-olosing
oonst* te tion^s the final number of stops will be a proportion
-c
26 of the stops wh*_ioh occur if they are omitted. Our problem
reduced therefore to th^e case in • hioh there are no closures.
It is, I ho e, also fairly clear thet the number of stoos will
iu>
not be sprreciebly effected by the lomSanmcfx brenoh arrangement
of the web, but only h_y the number of letters occurring in it.
These facts enable us to make 8 tabL e for the oeloulation of
th e number of sto^s in pny case where there is only one wgb.
The meth_pd of construction of the tp>ble is very tedious pnd
uninteresting, itods The t^ble is reproduced below
of letters
on v;eb
H-M factor
2
0,92
3
0.79
4
0.62
5
0.44
6
0.29
7
0.17
8
0.087
9
0.041
10
0.016
11
0.0060
12
0.0018
13
0.00045
14
0.000095
15
0.000016
16
0.0000023
4-C
No. of answers =26 x H-M factor
c is number of olosures
A similar table h as also been made to allow for two webs, with
To th e case of th^ree webs
up to five letters on the S' oond. fiEyBjid ■ ih±x it is not worth
wh ile 8nd hardly possible to go. Cne^canj^get a sufficiently
good estim te in such oases by using common-sense ineoualities ,
KXgxx&xsecsjJL-idxwElsT^Bf kh>rrKSiXiKttHi;si.Aacui7'Rxthir!i.xu£-<.LxdiAX J.3d
T ff d ffl gw wtrta r B -Trmntaffl r - o f yr m w-nt w rsr e.g. ifc we denot e the H-M f^otor
for the oase of webs with m,n,and p letters by H& H(m,n,p) v
shall have th e common sense inequalities
3 . 2 )
SlmTW
H(m, 3, 2)
> H(m, 4,0)
To see whet kind of contradictions are deteoted by the
machine we cn take the oioture, Fig end on it ’"rite
against eaoh letter an y Stecker values of th*-t letter whioh
can be deluded from the Stecker hypothesis which is read off
the spider when It stops, SJnondix This h^>es be n done in
mi/
Fdg for a case where th e input w r s on letter E of the
diegonal board, and th e relay R dosed when the machine
stopped} if the position of the ston were correct at ell the
oorrect Stecker would be given by the points of the diagonal
also
board which were connected to Er, and they will be the diredt
oonsequences of the Stecker hypothesis E/R,
±jrfci*rxx®xxhHil^xwi±hx*xx i**± nrmct iks^' ifljpc x ^xidtMstxwrtslalyxx
fcaxskie ' tBxde<ttt*ax®iixthaxafchx*rx*jiBQrj:9atxn-Rexx As we are
assuming th^et R was th e only re ley to close rtkxt hxa xg t j o T ,
this relay oannot h c ve been connected to any Eth of th e others,
or it would h eve behaved similarly. We oannot therefore deduce
any oth er Stecker value for E than R, and this explains why on
the 'main web' in Fig th e^e is only one penoil letter agianst
each ink letter. Wherever any penoil letter is the same as an
ink letter we are eble to miax write down another penoil letter
oorrespon ding totth e reciprocal Steoker or to the diagonal
connections of th e board. In one or two cases -e find that the
letter we migh t write down is th ere already. In others the
new letter is written against th® wn a letter of one oft the
minor webs; in such a o°se we aha clearly have a contradiction,
but as it does not result in a second set of pencil letters on
the main web the machine is not prevented from stopping. There
are other contradictions; e.g. we have Z/L,W/l, but as L does
not occur ltnth e crib this has no effect.
II s
V 7? T*
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ft 7L. Jk*it+x o*J+* y Xu U/U^ , ^vk ^ u~y*uU4 ^ XU
gjH w «v3A. ^ ^ *-u ~> *A, W A •'
7/Vj T/pjT/n K fl.-vA a-y*. k*i- v$+rU~-<~J- ^ /U ! /u~*^ ■
A
8 8 g ■ H-5- a =?
I 3 .3 U- 5 t> 1 8 q to il fi IS 1» ISlI, H IS to 2» 3.1 11 &MSf,3k
The machln e gun
When using the snider there is a gre«t de*l of work in
taking down data about stone from the machine and in testing
these out afterwards, making it hardly feasible to run cribs
which nunfl give more than 5 stons ner whe c l order, itxitxxxxi
direct consequences b txtXm of any Stecker hypothesis at any
position 8re already contained in th e connections of the
it should be nos^ible to make the machine do the testing itself.
It would not be pjecessary to improve on th_e storing arrangement
of tfcL^e spider itself, as one could jaxiexaaxii rxxagKweKi:nffh±jrtr
xb usejthe spider xtnxxno as already described, Knsjhpcrhan xavsrx
and have an errangemei\_jb by v/hioh, whenever it stopped a further
meohanism is brough t in to nley, which looks more closely into
th e Stecker, Such e mechanism will be described as a machine gun,
regardless of what its construction may ba.
that would be passed by fcfrg machine gun as possible would be
ratio
higher than th e ixsaartioH of ±®±*± spider stops to total
pos sible hypotheses. Consequently the amount of time that can
economically be allowed to the machine gun for examining a
position is vastly greater than can be allowed tojthe snider.
We might for instance run a crib which gives 100 snider stons
per wheel order, an^d the time for running, anart from time
spent during stops might be 25 minutes. If the machine gun Were
allowed 5 seconds per nosition, as coranared with the sniderSs
1/10 second only 8 minutes would be added to the time for the run.
As the complete data about the
8
IVO
When th e spider stona, nornrlly the points of the diagonal
supposedly
hoard which 8 re energised are those corresponding toJ,f«>lse
Stecker. Naturally it would be easier for the machine gun if
the points energised xax* corresponded to supposedly correst
Stecker. It is therefore neoes^ry to have some nrrangeinent
by which immediately after the spider stona the point of entry
to
of th e current is altered to th e point whioh the relay
which closed was connedted, or is left ur altered in th e oase
th at 25 relays closed. fiHKyrBMibiaxdaTiKsxfxcncdBingxtiaiixr
Mr Keen has invented some devioe for doing this, depending
entirely on relay wiring, I do not kno^- th e details at present,
buixxtrrmightxfesxMx^i^xxitkrTctrhiarxxistvasxraBXBXKxttaftxth*
W«xiraK±dTKrxHWgex*hxixTfrtkxBTinxKh±awxxiHrHXx , but aTyp Pr * n tly
the effect is that the machine does not stop at '■■11 except in
oases in ”'hioh ±4® either just one relay closes or 25 releya
close. In the case that 25 relays close the cu rrent is allowed
to oontinue to enter at the seme point, but f just one relay
closes the point of entry is ch anged over to thid relay. This
method has the possible disadvantage that a certain ndmber of
possible solutions may be missed through not being of normal
type. This will only be serioas in cases ’"here the frequency of
spider stops is very high indeed, e.g. 20£, and some oth er
method, suoh es ’Rings tellung cut-out’ is being usedfor further
reducing the stops. An alternative method is to have so-’e kind
of a* scanner’ whioh will look fir xhigfrjucxg. relays vrhieh
are not connected to any otherss . Which method is to be uded is
not yet decided.
At the next stage in the prooess we have to see whether
there are any cohtra dictions in the Steoker; in order to reduce
th e number of relays involved this is done in stages. In the f±
first stage we se v/hether or not there are two different Stecker
ve.lues for A, in the second wheth er there are two different
v-lues for B, and so on. To do this testing we have 26 relays
-**
A /f-Vf U* luX. U* UtkJL A) 14 W .
which pre wired up in such * wey th °t -e cnn distin-uidh
v;heth er oi not two or more of th em *>fe energised. When we ere
testing th e Steoker values of A we h ave the ?6 contacts of the
A line ofthe diagonal board connected to the corresponding
relays in this set.x*ijbc±fcxfc*±sx!bc*k±®r Whet is principally
locking is some devioe for oonneoting the rows of the diagonal
board successively to the set of relays. This fortune tely was
found in post-off ioe atenderd equipment; vn± the clicking nd>i3e
that this gadget makes when in operation gives the wh ole
apparatus its natae. If w*» xmcwtdxlir^ find no contradictions in
th e Stackers of any letter the whole position is posted ns food.
The machine is designed to print the position and th e Steoker
in such a case. Here again I do not know the exact method used,
but the following simple arrangement se^ms to give much the
same effeot, although perhaps it could not be made to work
quite fast enough. Th e Steoker **rn given by nrinrtiaap-
'.Then any
typin’ one letter in a column headed by the other, ix xatxodr
being
letter is tested for Stecker oontr n dictions th_e relays
corresponding tothe Stecker values of the letter close. V/e oan
arrange that these relays operate keys of the typewriter, hut
that in the case that the e is a contradiction this is prevented
no relay closes nothing is typed. The loans of the typewriter *x*ix
corresponding
special typed instead
and some symbol lrx-r i tiara shewing that the whole is wrong. When
o err lege
being examined sttaotges,
Addit ion" 1 gadgets
Besides th e spider and rap chine -gun a number of other
improvements scxxmBwxbsiji of the Borabe "re now being dinned.
We have already mentioned th"t it is oossible to use addition? 1
data about Stacker by connecting up points of th e diagonel
boerd. It is planned to m^ke this more streigh tfornard by
leading the points of the diagonal bfcerd to 325 points of a
plug board! the plug board also h^ns a greet many points all
connected together, end any Stacker whioh one believes to be
f"ls© one simoly oonn ots tojthis set.
Another gadget is designed to fled with t taxgxsftxy.EhK r x xt hy r g
sonix ceses such as that in which there are two 'webs* with six
8nd no chains
letters on eaoh. A sit little experiment will show th"t In the
gre- t majority of c»8es with sdch data, when the solution is
found, the Steoker value of a letter on either web will imoly
the whole set of Stackers for the letters of both webs : in
ax th6 ourrent terminology, "In the right place we oan nearly
always get from one web onto the other". If however we try to
run such data on the spider, even with the machine gun attachment,
there will be an enormous number of stons, and the vast majority
of the.se will be 0 "s-s in which * we have not got onto the
sodond web". If we °re nrepared to reject these possibilities
v/ ith out testing them we shall not very greatly decrees® the
probability of out finding the right solution, but v~ry greatly
reduce th e amount of testing to be done. If in addition the
spider oen be persuaded not to stop in these positions, the
spider time saved will be enormous. Some arran eaent of this
xx3s kind is being made but I will not ettemnt to describe
how it works*
With some of the ciphers there xaqrxbx is information about
the Ringstellung (Herivelismus) whioh makes certain stopping
pieces wrong in virtue of their oosit on, -nd not of the
alphabets produeed pt those nos it ions. There is an arrangement,
known as a ’Ringstellung cut-out" which will nrevent the machine
from stopping in such positions . The design of suoh a
cut -out dearly presents no difficulties of principle.
There are also plans for "majority vote" gadgets whioh
will enable one to make use of data which is not very relaible.
A hypothesis will only be re.-°rded a r re.ieoted if it
contradicts three (say) of the unreliable pieces of data.
This method may be eonlied to the case of unreliable data
about Stecker,
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1-u,
Chapter VII . The German Navel Enigma Cipher
Historical
In the pp^iod from ebo^t 1931 to April 30, 1937 the Naval
muoh
cipher liacdbcsni used the seme indicating system as the other
German service enigma ciphers, viz. the 'boxing 1 method
recommended by the firm that sold th e 'oommerdial' enigma.
With this ±n system as well as the set up of the machine
consisting of wheel order, Rlngstellung , n nd Stacker, there
was a window position fixed for the day, and known as the
'Grun date Hung' . Jpien it was desired to enoirher a message
from a JList of about 1700 trigrammes e.g. ZLE
one first ohose three letters at i^dom wxxxx&fiffi. 6ne then
set the meohine to th e Grunds tel lung and enciphered ZEEZEE.
The resulting six lettems were put at th e beginning of the
message, and the remainder , of the mess ge oon^sistet^f th e
result of enciphering the plaintext with pre-stprt window
position ZLE, (This differs from the other boxing indicating
systems in th at jdat most of these allow the trigrarame such as
ZLE to be ohosen 8t random instead of from thxw a restricted
list.
*The -eekness of this indicating system is that a great deal
of information is given away about th e ' Grunds tel lung ' . If
and a known diagonal
there were no Steokef , and the trrffio amounted to 100 messages
per diem it ’"ould be possible to find th e connections of the
meohine, and if there were Steoker but the connections of th e
machine were known it would be possible to" find the keys every
from
day wxtja the same amount of traffic. SxxBHSwxfBXxaxxmxtwxtiacfc
To explain the possibility of finding the keys let us suppose
that the following were a set of indicators for on e day's traffic
UJOOBL
VEYITM
ALAJMB
XDVBXV
QLYAMM
GRYLZM
JIPSWW
YEUTI
BMARFB
TZNGOR
AYIJPI
UJNOBR
OQBVCY
ENXDUJ
AFIJVI
MIEHWZ
KMIPFI
NUZIJG
IIFQWN
KGAPRB
SMLEBX
RFXCVJ
LGKZRP
BANRDR
GXJLEF
EKXXYJ
TXNGER
OIKVWP
TIOG'WL
LNVZUV
CZNYOR
RLFCMN
IMMZFC
MIDHWU
AUXJJJ
UWIOSI
IXRDEK
EYYXPM
ZCDWLU
EZS50H
VZPEOW
RSICAI
APTJNA
GJWLBS
CWTJYSQ,
RBDCQU
IHVQ.IV
OQJVCN
RZHCOO
AVRJKK
GIILWI
UAMODC
SCQKLE
DLYMTvC.l
EYVXF7
JNZSUG
UANODR
HALUDX
UOG OHT
ZFUWVQ
JSWSAS
ZDIWXI
KPBPNY
FTGD'-T
HEKUTP
UIT.V0 IS
In th © two indicators TJJOOBL and UANODR th e repetition of
th e first letters is followed hy a repetition of the fourth
letters. Th at this must always hepnen is clear from the
fact that th e fourth letter arises from the first, by
enciphering at the postition direotly after the Grundsteliung
and re -enciphering three nieces further in. lijoaxtharc
ratijssacaxs This phenomenonenebles us to tell very cuickly
with -ny oipher whether the Boxing form of indication is
being used. From th e indicators we c°n find the effeots of
th e three repeated encipherments. In Fig n . we have entered
tmcmsxsraaamMm in one of the columns against eeoh letter the
effeot of enciphering it first 8t the oosition Immediately
after the Grundstellung* and then et the position four places
after the Grundstellung: thus we have th e entry J against A, with
five
fsnsx dots. This means that A eno inhered at the first "nfi fourth
positions gives J, and th"t this information h as be^n given us
from six indicators, which are actually ALAJMB,AYIJPI,AFUVI,
APTJNA jAVRIKK^AnXJJJ, The other two columns give us the reaultn
of the encipherments et th e second and then the ihi fifth
position, and at the third and then the sixth. We get for the
result s of these double enoipherments
G 4 G 1
. . .JTOaMSKP.V .ISS0S. . .{DHffiJSEESBRCSI)
g b g b
(HTOJBqCUIFVKY) (OHBISAIKETGRZ)
Vi
(V ) ( 1 ) UTUKKPWSHODC ) . . .DUQEZGTABTMC . . .
G 4 G 1 here means the enoipherment with the first alphabet an d then
with the fourth, the reversal of the natural order being in
agreement with mathematical tradition. Thera can be no
doubt as to h ow the substitution GgG^ Is to be completed, but
et first sight it might appe-r that there are two ooseibllities
for G^Gjl . However if we remember whet we found out in the
section 'alphabets and bores' via sen that it must be possible
to peir off the cycles of G^ into ones of equal length.
'-here ere various things which could he done now. Of course
onemight put the whole data onto the spider, hut at th e time
that this system was in force no such machine had he^n thought
of. Another method, which vws • hat urine i^ally used hy the
Poles is to h«ve a permanent catalogue of the box shares xfctxfcx
for G 4 0^, GgGg, GgGg for every Grunds tel lung, assuming ^het
there is not a T.O. between the first ^nd Ip si of the six
elphahets. txrtlr-sse feax x x h xTJff s xKgK- ri b t A (MMt MDctx«±xthacxatag)cac^
or numbers
If v/e give some standard order to the hox shares, we o^n also
put the possible series of three hox shares into an order, end
can enter against each set of three hox shapes th e Grundstellungen
for which th is set is realised. To use this oetelogue with
our problem -e should work out the hox shapes viz. G4 Gj is 26,
G5G2 is 86, G^G 3 is 24,2. Them hox shores scctua Hyrfaxxextfcexx
XHndnExsx±2ptT 26 actually has the nuher 1 and 24,2 the number 2:
they 're the two commonest shines «s can he seen f^om the table
p . We then look ur 1,1,2 in our catalogue, and find
about lfO entries against it for each whealorder. Ea«h of th ese
will have to he tested art in some v, py or other. The most
satisfactory method se--ms to he thidq We form th e rermutation
4*8 4^3834* 0 6 G 3 G 5 0 2 , It is
( PBGKHNliD JYWHISH&UJIV ) ( CXZL ) ( 0 ) ( T )
so th at this permutation is of the class 20,4,1,1. For eaoh
possible rundstellun it is possible to celoulete tkix the
corresponding class for the unsteckered alphabets. This can
fortune tel 1 ' he done mechanically by means of a form of
r cyolometer f , It would he as "'ell to enter against each position
the c lass of this rermutation, and this might h«ve have he-=n done
et th e time of oon struction of the catalogue. Aartxxilyxlk®
In the oase in question the right Orundstellung is found to have
th e position 1,1,26 with wheel order ixi I, II, III (servire
machine, Umkeftrwalz A), The corresponding boxes are
'33
DT PZ CE
MG RN TQ
HL UG XU
UP JL JD
CK BE FO
AI QX RM
EJ OD NY
XV H KB
BQ MI PV
NS FW WL
OR AS IG
YVf KH HZ
FZ YC Is
We wrtxrHjtiyx must h*ve V/A or V/S. If V/A we can identify the
cycle (OHIWSADXRZTGRZ) of the G 5 G 2 with stecker, with the
half compartment of the second box in this way
(OHIWSADXETGRZ)
(CHSWIVDXELGNZ)
i.e. we g ratra h've to ps^.t the stecker 0/C,l/S,A/V,T/L,R/N, and
that H?W ,D,X,E,G,Z ere unsteck' red, This large number of
unsteckered letters is a strong confirmation, and the repetition
of the Stacker I/S is further confirmation, Y/hen we fit the
res of the box 1 3 together wa find tha + these five *re eii the
Stecker.
There ere other methods th**t can be am-died, defending ^nthe
number of Steck6r being smpll. The number of Stecker Ased in the
Neval was 6 from 1931 to Nov. 1938 ^nd uoseibiy later. We night
for instenoe have assumed that A c nd S were both unsteckered
A
and therefore assumed that the constetation S occurred in both
the alphabets G 3 and G g , With the Turing sheRts we could find
the possible rositions for this, andthen use * cyolometer to
test tirade the box shapes in those positions. This is naturally
only --orth while if we have no box-shape catalogue. Another
possibility is to Equate* the boxes, i.e. to find out from the
permutations G 4 G 1 ete what the original xwrmut alphabets G^end G^
were. In our c p se there *» r p aotual y 13 different possibilities
for G lt 13 for G ? and 12 for G 3 . There are two things we can
do to distinguish between the correct and the incorrect
possibilities. We c*n use known statistics about the list of
admissible message settings, ohoosing that combination of
alphabets that gives the aurxt greatest number of- moooa? * o
0 0 t tings t h air he vo hs o c curred nrr
repetitions between the message settings for the day in question
and message settings of previously solved d»ys. We mighjt “ljgQ
do r ’Banburismus' i.e, we right make use of the* * f"ct thet
if two messages are written out with letters that were
enciphered at the seme position written in the seme column, then
the number of iwttarx repetitions of letters in « column will
be the seme es if the messages had not bean enciphered, and
otherwise pleoed. Aotually this efiect ve* very smell for
the Naval traffic in 1937 and e rlier. The repetition frequency
was 1/20, es compared with 1/16.5 for the 1940 "aval traffic
end the Air traffio, *mt l/l2 for plain language German, and
1/86 for incorrectly placed enoioftered messegesithe repetition
frequency is the r"tio of the number of identical pairs in k the
columnsto the number of pairs in columns, identic"! or not).
With so low a repetition frequency it is zix**tx±isx***z
extremely difficult to t equate the boxes unless the traffic is
rather heavy. This method however applies quite well with
the Air traffic up to Sept 14 1938, but there tfce^e are better
methods of equating. Onoe the boxes have been equated by one means
or another we shell have many more cases of ixa half-bombes
which we can assume to have been unsteckered. This method
will nearly always get the result, if the equating oan be done.
After we have found the rod position of the Grundstel lung
and the Stacker it only rf-mains to find th_J» Ringstellung.
Usually this would be known already, as, at this period, the
wheel order and RJLngstellurtg were only changed -bout once
a fortnight. However if these have ? ust. be^n 0 hanged it is
n ecessafry to read one message. This could always be done, as
a great many messages wer sent in two or more parts. In
such 083es the call signs and signatures of the parts were
essentially the s^rae, and the Knmfc nerds efter the first
began by say^JLng thet they were continuations, giving * the
time group of the iiBit n a message as a reference. Stabsxxa jcx dsxa
lsxtta The method of giving numbers at that time was to use
on average
therefore will
been
total
last part of the
the tor> row of the key board end P thus
QJNERTZUIGP
1234567890
The number was put between Y's to shew that it was «= number,
end the whole repeated es e check. The continuation of a
message whose time group w eg 2330 would begin
FORTYY/EEPYYWEEPY , We could then find th e position where this
raessnge started by single wheel processes, «nd es we already
know the window position of the start, we oan calculate the
Rings te llung.
On the 1st May 1987 a new ind io a tii^ system was introduced.
The first two groups (four letters eeoh) of the message were
repeated at the end. This clearly showed that these two groups
formed th e indicator. The repetition also showed that no
check oould be expected within the ±n first two groups
themselves. This w r s discouraging, »s the essential weakness
of the boxing method was that the s^n^hing was enciphered
twioe with the machine. With the new method of indicating,
whatever it is, th e best one o r n hope is that either it
will enable us to’ set’ the messages, or thpt war from some
information about the setting of the messages obtained
elsewhere we may be able to deduoe something about the ecftcxx
machine setting. However the first thing to be done was to
find out how the indicators worked, and if wa neoesssry
therefore to try and read some messages with which the new
system was being used. To do this one can use the FORTYWEEFY
messages, and apnly one of the methods described at the
beginning d>f the last chapter. In this wpy the Poles found
the keys for the 8th of May 1937, and as they found that the
wheel order and the turnovers were the earae as for the end
of April they rightly assumed that th e wheel order end
Ringstellung had remained the same during the end of AJ>ril
and the beginning of May. This made it e-sier for them to
find th e keys for other days th e beginning of Mpy end
th ey actually found the Stecker for q ck a a c t the 2nd, 3rd, 4th, 5th
end 8th, end reed shout 100 messages, ilxxxxxf sxKdy iiax txi xxscli
iraxxsr The in dioators en d window positions of four (seleoted)
mess ges for th e 5th were
Indies tor Window stert
KFJX EWTW P C V
SIXG EWUE B Z V
JMHO UVQG
JMFE FEVC
MEM
W- Z K
The repetition of the EW combined with th e repetition of V
suggests that the thixdxxndy fourth fifth end sixth letters
describe the third letter of the ' indo- position, end similarly
one i^ ledto believe thet the first t- o letters ofthe indicator
represent the first letter of the window position, end thet the
third °nd fourth represent the second. fcfcixxHffstrtx
±3 ucgxB ?resumebly this effect is somehow produced by weans of e
tebl :■ of bigrem’ e equivalents oflette-s, but it oennot be ^
done simply by replacing the letters of the window position / ^
■ ith one of th eir bigram- :e ecu v*lents, and then putting in
a dummy bigramme, for in this o«se the window position
corresponding to JMEE EEVC would h«ve to be say MYY instead of
MYK. Probably some encinherment is involved somewhere. The two
most natural alte r n p tiv r s are . ±$ i) The letters of the window
position e-e replaced by some bigrqmme equiv lemts and then
the whole enciphered at sdme ’Grundstellung’ , or ii) The window
position is enciphered at the Grunds tel lung, and the resulting
letters replaced by bigramme equivalents. The second tof these
alternatives was made far more probable by the following indicators
occurring on the 2nd May
EXDP I¥J0 ¥ C P
XXEX JXJY TV E
RCXX JLWA. NUM
With this sec And alternative we can tfmctdK deduce from the
,3 7
first two indie tors that the bigrammes EX and XX have the s'-me
value, and this is confirmed from the second and third, where
XX and EX oo'ur in the seoond position instead ofth e first.
It so happened that the ohange of indicatin'' system had
not been very well made, and a oertnin torpedo boat, wlththe
call sign AFA: hed not been provided with the bigramme
tables. This boat sent a message in -nother cipher explaining
this on the 1st May, and it -os arranged th°t troffio with
AFA: was to take piece according tothe old system until May 4,
when the bigramae tobies would be supplied. Sufficient traffio
passed on May 2,3 fax to end from AFA: for the Grundstellung
used to be found, the Steoker having alre dy been found from
the FORTYWEEPY messages. It was natural to assume that the
Grundstellung used by ABA: we 3 the Grundstellung to be used
with the oorreot method of indication, and as sson os we
notioed the two indicators mentioned above we tried this out
and found it to be th e cose.
There actually turned out to be some more complications,
at least
There were two Grundstellungen instead of one. One of them was
called the AlLgeineine end the other the Offiziere Grundstellung.
This made it extremely difficult to find eithter Grundstellung.
The Boles pointed out another possiblity, viz th»t th e
trixgranries were still prob r bly not ohosen at random. They
suggested that probably the window positions enciphered at the
Grundstellung, rather than the window positions th emselves
were taken off the restricted list.
a i Nov. 1939 a prison er told us that the German Navy had new
given up writing numbers with Y. ..YY. ,.Y and that tiaxKyxjrarra
the digits of th e numbers were spelt out in full. When we heard
this we examined the messages toward th e end of 1937 which
were expeoted to be oontinuati ns 'n d wrote th e expected
beginxiinings under them. The pro 'ortoin of ’crashes* i.e. of
letters apparently left unaltered by encipherment, xfcxx then shews |
how nearly correct our guesses were. Assuming that the change xra
t2P
mentioned by the prisoner hed elready taken piece we found th et
ebout 70% of these cribs must have been right. Further ’crash
analyses ’were made for other periods up to Aug 1939, all with
fairly favourable results. At the same time there had been some
chan ges in the meohine, known toh eve taken place because o^the
corresponding changes in the machine used by the army and air
whose treffio h ad been read. In the summer of 1937 the UmkJjtdrwalze
had been changed from A to B, and in (^Deo? 1938 two new wheels
tr IV and V had been introduced, iHxxttaLaaJuusxxjcxxixxjEa After U-t
i uj* k- U ** 1
the beginning of the war (Sept 1939) the FORTYWEEFY messages &
v.
were no longer traceable, because there were no more 0*11 signs, i
of thid kind ^
However there had been some traffio at various times during
manoeuvres and arises sinde the occupation of Austria, Thaxxx
xsvmedxtoxb
and there were a few days where there
was both traffio with and without call signs. We h6ped that
we might be able to find the keys for some such days endfco so
to find the kind of thing that was said in the traffic without
call signs, There seamed to be
some doubt as to the feasibility of thid ol«n, "s itxxnxuejLiyx
the oall signs traffic on any day was always either the whole of
the Baltic traffio or the whole of the non-Baltic traffic, and
the Baltic traf - io in 1937 usedto be on a differen t key from th e
rest. Following this programme we found the kevs for Nov, 28
1938 and for a number of d-ys near there. The number of Stecker
w s 6, Th e wheel order end Ringstellung seems' - to remain constant
for ebout a week; at an y rate they did not change between Nov 24
and Nov29. The Stacker fr gim gkxxxxPkis xxxy were nbt hatted; the
ssme letter wan. never steckered on t o oonsecutive day3. This
of course might be extremely valuable. If the traffic had been
h eavier it v/ould have enabled us to find th e keys so long as
this lasted, and there were 'ny cribs. Actually we got no
furth er than this* es at this point a good deal of data was.
'*>
'pinched’ fr m a German boat, enabling us get the keys for
April 22-27 194C. At the same time we pinched a book of
instru ctions telling us th e precise form of the indicating
system.
To enoipher a me s " a ge the operator ch ooses two trigrem^es
ou t of e book. The first of these trigrammes is called the
•Sohluesselkenngrup 'e' . The ohoice of thi3 is partly determined
by the nature of th e message: e.g. ill ’dummy’ mes^a es have
th e Sohluessel keengrupre token from one port of the book and
genuine messages &ave them token fr m elsewhere; we do not
kn dm very much about th ese. Th e second trigramre isoalled
the Verfeharenkenngruppe , Suppose the Sehlues^elkenngrupoe is
CIV and the Verfahren kenngrup ^e is TOD then the oper-tor
ohooses two dummhy letters, Q «nd X say, and writes this down
Q C I V
T 0 D X
FSom the Verfahren kenngruppe is obtained the window position
for th e start of th e message, by eh ciphering at the Grundstellung.
From th e eight letters above, one also obt'ins the indicator
for the messoge, by substitution from e table which gives
bigfarame for blgrarnme. Th e xsxii* substitution is done by replacing
the vertical pairs above with bigram ies, e.g. ±£x in this oase,
if the substitute for C^T were DA, and TH for CO, PO for ID, «nd
CN for VX then the indicator for the message is DATH POCN,
Apart from the Schluessel kenngruppe feature this is the
method we had inferred w«>s being used, ihs^fthana This extra
feature accounts for the bigram es in th e indicators being
almost perfectly h atted. Also th e fact that it is never the
message setting itself wioh Is ohosen at random by the operator
eliminates any remainin'’ hope th at one might use ’operator’s
psychology’ to help in finding out th e alphabets. From our
point of view of course the Sohluesael keangrupren raigh t
as wel‘ not exist, an d th e tbigram e lists’to us remain
letter entered
Foss sheets with on e surtrxy in eaoh sou are, and not two .
There is however the restriction that there must be exeotly 26
occurrences of eaoh letter.
Methods of ree d ins the in d ividual messages
With th e system of indication that h as been used sinoe
‘•iay 1937 we are not able to reed all the messages as soon as
we have read on e. A few may be re^d by single wheel processes,
starting from a short crib, bu t we oermot hone to read the
whole traffic in this way. Also, when we have found the
Grundstellung xr xxs x xfc hx a fci*x±B , an d if the-'-e is plenty of traffic,
we may be able to make use of tkxx some bigrarnmes whioh oo^urred
in messages already read. These methods »re not enough by themselves/
In th e 1937 traffic there was no 'mot probable* , and we had
planned a method for finding the right starting position, making
use of the fact that th e correct deCode would probably have
more letters E in it then any of the others. It was intended
to have a long punched paper roll, th e punching shewing the
effect of enciphering E in the various positions. This paper
was to move under a series of about 200 brushes whose position
was determined by th e lett rs of tie enciphered message. The
number of brushes which poked through the holes at anv moment
was th e number of letters E in th e decode of the message, the
window
position sixth® bein g determined by the position of the roll.
All positions giving more than * ce^t'-in number of letters E
were to be reoorded and these positions indenen dently tested.
This nr chine w s called *th e rack’.
It was never n ecessary to make a raok beo-use when the
1938 messages were reed it was fibund th at thexac word EINS
occurred veryfrequen tly. We therefore made a catalogue of
th e encoded v lues of EINS st every possible starting position,
an d arranged the enoodcd values in alphabetical order. The
un^nalysed catalogue was nade by enciphering first E at every
possible position, then I,N and S, This was done with the
automatic typewriting en ifenas. The values of I were stuck below
the values of E with a stagger: the values of N and S were
underneath th ese again, with siitable. staggers , The esult was
that the effect of enciphering EINS a one a red in vertical oolumns.
This unaaalysed catalogs was known to th e girls as ’oorsets*.
In analysing the catalogue we took 25 sheets n amed A to 8, with
S omitted: eaoh sheet had 25 lines, named A to Z with I omitted.
Supnosing on she t 13 and line 4 of th e corsets we found
OM
LVOM as a value of EINS we would enter 13.4 on line V of sheet X,.
Xxmlxtiir* In a later form of the oetalogue also made
Existence sheets*. In the existence sheets we would enter M
in line V and oolumn 0 of sheet L. To use the catalogue w«
first analysed the tetrsgraiames in th e messages irtcxikx
accoring tot he if first letters. One would then take th e fi
existence sheet and go through all th e messages marking the
tetragraimes which oo^urred on th e existence sheet, °nd marking
against them the entry (e,g. 13,4) from the catalogue. Afterwards
one wouJ.d have to go beck to the corsfcts, end search in the
right line for the tetregranme, and work out ita position:
this wag done with a oardboard strin end kniwn 83* snaking*.
Having foun d the position one would have to set up th e m chine,
decipher the tetragranne, verifying that it g n ve EINS end them
continue to decipher and se- if one eontinuedto get. sense,
This process has since be r 'n greatl ,r improved. Instead of
making the corsets off the'X-raechines *ye have a machine oalled
the * test-plate* or 'baby* which tyned out the results of
enoipherin g EINS in all positions in a much more convenient foun.
Also we xa&JAn n o longer analyse the groups by hand, but h°ve
together with their nos it ion
them punohed on cards , which are
then sorted into alphabetical order, and listed, A further
improvement is th at the test-plate is now made to punch the
cards directly.
Roughly, our programme when the wheel order, Ringstellung,
and Stecker for a dap- h eve been found, is as follows. We make
it
en EINS catalogue, and use to get out nairs of messages in
whioh the second indicator bigram- ’e of one is the same °s the
third indicator bigramme of th e other. If we have four suoh
oases we have sufficient data about th e Grundstellung to be
able to find it by means of th e Borabe, provided that we have
found th © double T.O. i v/e then continue to get messages out
with the EINS catalogue; each message gives us some tet values
of bigrammes, which are entered on a Boss sheet. JErom time
to time w® go th rough the messages substituting for the
recently
bigrarames the values that have been found from the messages,
v/ith messages for which we know the values of t'-'o of the
bigrnmmes we apply th e method known as ♦twiddling* or ’bonking* .
We heve to deoipher the first few letters of the message at all
of the 26 plaoes consistent v/ith our knowledge of the bigr®mmes.
This is usually done in column*, one column at a time,eaoh
column corresponding to a letter of the message. The twiddling
is best don e on the Letohworth enigmas, as they h eve no
automatic T.O, Some more messages can be solved by when on®
bigramme is known, preferably that corresponding to the L.H.W,
on th e test-plate
by deciphering a few letters at every one of the 676 plaoes. But
this method is rather difficult to v/ork in oreotice. It seems
rauoh more difficult to spot the right answer when one h"s to
look through so many possibilities. The right answer is hardly
ever noticed unless it is onfe of the obviousones such as
BIENEfWESPE ,MUECKE ,MOSKITO ,HOKNISSE ,KRKR ,ANAN ,ADMX ,GRUPPE .BfiSWJ ,
The case "here the R.H.W. bigr°mme is known cannot be done on
th e test-plate at all. One ataqrxtes: can of course use the X-maohinas
in muoh the same way as was done with th e original form of
EINS catalogue. This has never beeh 8 success, Onr can also
use *hand methods. On e can go through the message looking for
places wh ere two consecutive letters occur on th e saraerod.
The deciphered values also oocur on the same rod, and we can
examine the rods for possible bigreranes, Combining this v/ith
th^e Turing she ts, Kendriok has solved quite a number of mes ages.
This method is known as ’olioks on the rods’.
■Ji lcS k r IUS uUwu. /U /£-
^ (1 ^ ft* k.- Cl H f ^ U **,/*,' s ^-
SUZ.
Idoatlfloatlonof blgranae lists and ef valuation of wrinamm ^grng s.
The T ehfahrenkenngruppe (T.K.O. or trlgranine) Is as we hare explained
not choeen at random , tut from a Hat of a out 11 , 000 , and within thla llat
the eholaei are not made at randan uniformly . This fact enables us to
Identify which hi gramme llata are being used , for li we chooae the right
bl gramme llat and work out the T.K.O. we sha}l find that a comparatively large
proportion of the m have ocourred bdfore, and If we chooae the wrong one ,
a comparatively email proportion.
The more preoiee theory of thla Identification^ aa follows. let ue
auppoae that o lithe different trlgranmes <*-, have tka been uaed
once
before xtlnu, twloe eto. Iwt ua oalj. a trigramme whlofl has
occurred before ^ t femes a trigramme of the h -claas . W e can them expreaa
our information in the form:
Of the oocurrrenoea of trlgramnee there have been In the / • olaas,
la the 1 - cdaae, in the i - class etc:
®ow take a random eample of these occurrences, forming a proportion <*. of
the whole, and let ua imagine that this random sample oonalata of the last
of the trl gramme a whloh wera found. There will be oiu. f In the 1 class,.
in the 2 olaaa, eto. ®ow the ones In the 1 class would have been , when they
were fpund,onea whloh had not ooourred before, and those which In the 2 class
ones which had ooourred before onoe, and eo on. Hence we can aay that for the las at
! occurrences of trigramme a entered, the numbers s whloh had
ooccurred beforeyonoe,twioe,threetimes, . . . are in the ratios of n, ,3* ,
W e must expeot these ration to hold also of the next few occurrrencea to be
entered. The process of finding new occurrences of tri grammes and lookinf gup th
numbers of previous occurrences can therefore bo regarded ad like having an
urn containing cards, each of whloh bears a trlgramne|knd a number, and making
drawefrom the urn. The nujmber of oarde bearing the nukber ^ Is th be
proportional to (_•’■■*') ^ ^ ^ .On the other hand we have to consider the process of ,c
choosing trl grannie b at random. This 1 s to be xmgsxtstxamx oon^ared whth
kw
drawing cards from an urn containing oards in di ff ere n t proportions,
(
'Hu ^tpcess Ia , ^TLj2— per^tc/AJT" li^-yP fl£~ -#C«_
4o_^> er£— Crf'luiiv'jvS on. -^-i2__ ce-irJere— p -o_^ g -S o& — -^-Q — /Cr. A-o^ri^L 7 Lxdr~
^■^+WU2*v Lo-ejr^— yv^Mj/Il- <a ^JJ CCS, c£ — Ui~CtS
U£<e-dl , CtwJ^ *\jrt" 4*> N-e_- vk&c^ tt— * ^juS> -feiCc- t-e^-e^cA" 'rcjc^- o%-
ld^& — K*-CJ^ -kr^(Tn\^ 1^. — Ickjs>-uz-t^ Aip /v ai^ A-e-e^.
j I, , ' P*^ ^ V^&fl CeUotU~-j
CJrcU^-Oj^ ,u *Aj£_ ^ 1<^- Acn^ ^C^^julrufi-^. *U ' ^
- uu wiic* #.
-$u2_ KV^r»v -
<f£L>^ *A Jr*
CA-tU
J] 0^
Jach /trlgraianme must oocur squally of ton An this urn, and must of course hare
with It the number of previous occurrences of this tfcl gramme . *csr Imagine that we have
worked out a certain number of T.K.O. using a given bi gramme table, anf that we have
found out how many times each of them had occurred before. This can be con^zred with
being glvewn one of the urns, and told It Is ^:i on this being the random urn ,
and then drawing a certain number of cards from the urn. After the draw we have a
new Idea of the odds that the urn Is the random urn, and we should have a correspond! n
modified Idea of the odds that the btgramme list is the right one. Let us suppose that
the trigrammes, In the order as they were fraud* worked out, had the numbers
,r y. r s of previous occurrences, and that ro orre sp ond 1 ngly the cards
drawn frok the urn bore the numbers 0 ( . The proportion of oases of
draws of s cards from the urn , giving these results with the same order , is
u where k r la the proportion of r -cards in the urn.
Likewise the proportionof cases where this happenswlth the other urn Is
u * u. ' with a correspond ong meaning far • Then tfco odds on the
’I
urn not being the random one after the draw experiment are
C V,
In other words the drawing of a card with the number * m iraprovee the odds
^‘ f “ torof 1^/ ,B ^ to
- ’-o w hen it Is —
exocept in the oase
The same method may be applied for the Identification of some unknown bl grammes
By taking Into account a number od days traffic all using the asms bl gramme
table we my fing a number of indicatoras whose T.K.O. would be completely
known if we knew the value of a certain bigramne. If we make the right hypothesis
as to the value, we should get tr’ grannies agreeing with the etstistlos as before.
In this sort of oase, as the data is liable to be very scanty, It Is essential to
use the accurate theory as described above.
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